Trigonometry Examples

$\sec (\arcsin (\frac{x}{\sqrt{x^{2} + 49}}))$

Step 1

Interchange the variables.

$x = \sec (\arcsin (\frac{y}{\sqrt{y^{2} + 49}}))$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sec (\arcsin (\frac{y}{\sqrt{y^{2} + 49}})) = x$ .

$\sec (\arcsin (\frac{y}{\sqrt{y^{2} + 49}})) = x$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{y}{\sqrt{y^{2} + 49}})}^{2}}, \frac{y}{\sqrt{y^{2} + 49}})$ , $(\sqrt{1^{2} - {(\frac{y}{\sqrt{y^{2} + 49}})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{y}{\sqrt{y^{2} + 49}})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{y}{\sqrt{y^{2} + 49}})}^{2}}, \frac{y}{\sqrt{y^{2} + 49}})$ . Therefore, $\sec (\arcsin (\frac{y}{\sqrt{y^{2} + 49}}))$ is $\frac{1}{\sqrt{1 - {(\frac{y}{\sqrt{y^{2} + 49}})}^{2}}}$ .

$\frac{1}{\sqrt{1 - {(\frac{y}{\sqrt{y^{2} + 49}})}^{2}}} = x$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{1}{\sqrt{1^{2} - {(\frac{y}{\sqrt{y^{2} + 49}})}^{2}}} = x$

Step 2.2.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{y}{\sqrt{y^{2} + 49}}$ .

$\frac{1}{\sqrt{(1 + \frac{y}{\sqrt{y^{2} + 49}}) (1 - \frac{y}{\sqrt{y^{2} + 49}})}} = x$

Step 2.2.2.3

Simplify.

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Step 2.2.2.3.1

Write $1$ as a fraction with a common denominator.

$\frac{1}{\sqrt{(\frac{\sqrt{y^{2} + 49}}{\sqrt{y^{2} + 49}} + \frac{y}{\sqrt{y^{2} + 49}}) (1 - \frac{y}{\sqrt{y^{2} + 49}})}} = x$

Step 2.2.2.3.2

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y}{\sqrt{y^{2} + 49}})}} = x$

Step 2.2.2.3.3

Multiply $\frac{y}{\sqrt{y^{2} + 49}}$ by $\frac{\sqrt{y^{2} + 49}}{\sqrt{y^{2} + 49}}$ .

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - (\frac{y}{\sqrt{y^{2} + 49}} \cdot \frac{\sqrt{y^{2} + 49}}{\sqrt{y^{2} + 49}}))}} = x$

Step 2.2.2.3.4

Combine and simplify the denominator.

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Step 2.2.2.3.4.1

Multiply $\frac{y}{\sqrt{y^{2} + 49}}$ by $\frac{\sqrt{y^{2} + 49}}{\sqrt{y^{2} + 49}}$ .

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{\sqrt{y^{2} + 49} \sqrt{y^{2} + 49}})}} = x$

Step 2.2.2.3.4.2

Raise $\sqrt{y^{2} + 49}$ to the power of $1$ .

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{{\sqrt{y^{2} + 49}}^{1} \sqrt{y^{2} + 49}})}} = x$

Step 2.2.2.3.4.3

Raise $\sqrt{y^{2} + 49}$ to the power of $1$ .

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{{\sqrt{y^{2} + 49}}^{1} {\sqrt{y^{2} + 49}}^{1}})}} = x$

Step 2.2.2.3.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{{\sqrt{y^{2} + 49}}^{1 + 1}})}} = x$

Step 2.2.2.3.4.5

Add $1$ and $1$ .

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{{\sqrt{y^{2} + 49}}^{2}})}} = x$

Step 2.2.2.3.4.6

Rewrite ${\sqrt{y^{2} + 49}}^{2}$ as $y^{2} + 49$ .

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Step 2.2.2.3.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 49}$ as ${(y^{2} + 49)}^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{{({(y^{2} + 49)}^{\frac{1}{2}})}^{2}})}} = x$

Step 2.2.2.3.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{{(y^{2} + 49)}^{\frac{1}{2} \cdot 2}})}} = x$

Step 2.2.2.3.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{{(y^{2} + 49)}^{\frac{2}{2}}})}} = x$

Step 2.2.2.3.4.6.4

Cancel the common factor of $2$ .

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Step 2.2.2.3.4.6.4.1

Cancel the common factor.

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{{(y^{2} + 49)}^{\frac{2}{2}}})}} = x$

Step 2.2.2.3.4.6.4.2

Rewrite the expression.

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{{(y^{2} + 49)}^{1}})}} = x$

Step 2.2.2.3.4.6.5

Simplify.

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (1 - \frac{y \sqrt{y^{2} + 49}}{y^{2} + 49})}} = x$

Step 2.2.2.3.5

Write $1$ as a fraction with a common denominator.

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} (\frac{y^{2} + 49}{y^{2} + 49} - \frac{y \sqrt{y^{2} + 49}}{y^{2} + 49})}} = x$

Step 2.2.2.3.6

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.4

Multiply $\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}}$ by $\frac{\sqrt{y^{2} + 49}}{\sqrt{y^{2} + 49}}$ .

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}} \cdot \frac{\sqrt{y^{2} + 49}}{\sqrt{y^{2} + 49}} \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5

Combine and simplify the denominator.

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Step 2.2.2.5.1

Multiply $\frac{\sqrt{y^{2} + 49} + y}{\sqrt{y^{2} + 49}}$ by $\frac{\sqrt{y^{2} + 49}}{\sqrt{y^{2} + 49}}$ .

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{\sqrt{y^{2} + 49} \sqrt{y^{2} + 49}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.2

Raise $\sqrt{y^{2} + 49}$ to the power of $1$ .

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{{\sqrt{y^{2} + 49}}^{1} \sqrt{y^{2} + 49}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.3

Raise $\sqrt{y^{2} + 49}$ to the power of $1$ .

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{{\sqrt{y^{2} + 49}}^{1} {\sqrt{y^{2} + 49}}^{1}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{{\sqrt{y^{2} + 49}}^{1 + 1}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.5

Add $1$ and $1$ .

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{{\sqrt{y^{2} + 49}}^{2}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.6

Rewrite ${\sqrt{y^{2} + 49}}^{2}$ as $y^{2} + 49$ .

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Step 2.2.2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 49}$ as ${(y^{2} + 49)}^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{{({(y^{2} + 49)}^{\frac{1}{2}})}^{2}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{{(y^{2} + 49)}^{\frac{1}{2} \cdot 2}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{{(y^{2} + 49)}^{\frac{2}{2}}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.6.4

Cancel the common factor of $2$ .

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Step 2.2.2.5.6.4.1

Cancel the common factor.

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{{(y^{2} + 49)}^{\frac{2}{2}}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.6.4.2

Rewrite the expression.

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{{(y^{2} + 49)}^{1}} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.5.6.5

Simplify.

$\frac{1}{\sqrt{\frac{(\sqrt{y^{2} + 49} + y) \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.6

Apply the distributive property.

$\frac{1}{\sqrt{\frac{\sqrt{y^{2} + 49} \sqrt{y^{2} + 49} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.7

Multiply $\sqrt{y^{2} + 49} \sqrt{y^{2} + 49}$ .

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Step 2.2.2.7.1

Raise $\sqrt{y^{2} + 49}$ to the power of $1$ .

$\frac{1}{\sqrt{\frac{{\sqrt{y^{2} + 49}}^{1} \sqrt{y^{2} + 49} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.7.2

Raise $\sqrt{y^{2} + 49}$ to the power of $1$ .

$\frac{1}{\sqrt{\frac{{\sqrt{y^{2} + 49}}^{1} {\sqrt{y^{2} + 49}}^{1} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{\sqrt{\frac{{\sqrt{y^{2} + 49}}^{1 + 1} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.7.4

Add $1$ and $1$ .

$\frac{1}{\sqrt{\frac{{\sqrt{y^{2} + 49}}^{2} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.8

Rewrite ${\sqrt{y^{2} + 49}}^{2}$ as $y^{2} + 49$ .

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Step 2.2.2.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 49}$ as ${(y^{2} + 49)}^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{\frac{{({(y^{2} + 49)}^{\frac{1}{2}})}^{2} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\sqrt{\frac{{(y^{2} + 49)}^{\frac{1}{2} \cdot 2} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.8.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{1}{\sqrt{\frac{{(y^{2} + 49)}^{\frac{2}{2}} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.8.4

Cancel the common factor of $2$ .

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Step 2.2.2.8.4.1

Cancel the common factor.

$\frac{1}{\sqrt{\frac{{(y^{2} + 49)}^{\frac{2}{2}} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.8.4.2

Rewrite the expression.

$\frac{1}{\sqrt{\frac{{(y^{2} + 49)}^{1} + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.8.5

Simplify.

$\frac{1}{\sqrt{\frac{y^{2} + 49 + y \sqrt{y^{2} + 49}}{y^{2} + 49} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}}} = x$

Step 2.2.2.9

Multiply $\frac{y^{2} + 49 + y \sqrt{y^{2} + 49}}{y^{2} + 49}$ by $\frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49}$ .

$\frac{1}{\sqrt{\frac{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}{(y^{2} + 49) (y^{2} + 49)}}} = x$

Step 2.2.2.10

Simplify the denominator.

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Step 2.2.2.10.1

Raise $y^{2} + 49$ to the power of $1$ .

$\frac{1}{\sqrt{\frac{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}{{(y^{2} + 49)}^{1} (y^{2} + 49)}}} = x$

Step 2.2.2.10.2

Raise $y^{2} + 49$ to the power of $1$ .

$\frac{1}{\sqrt{\frac{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}{{(y^{2} + 49)}^{1} {(y^{2} + 49)}^{1}}}} = x$

Step 2.2.2.10.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{\sqrt{\frac{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}{{(y^{2} + 49)}^{1 + 1}}}} = x$

Step 2.2.2.10.4

Add $1$ and $1$ .

$\frac{1}{\sqrt{\frac{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}{{(y^{2} + 49)}^{2}}}} = x$

Step 2.2.2.11

Rewrite $\frac{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}{{(y^{2} + 49)}^{2}}$ as ${(\frac{1}{y^{2} + 49})}^{2} ((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))$ .

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Step 2.2.2.11.1

Factor the perfect power $1^{2}$ out of $(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})$ .

$\frac{1}{\sqrt{\frac{1^{2} ((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}{{(y^{2} + 49)}^{2}}}} = x$

Step 2.2.2.11.2

Factor the perfect power ${(y^{2} + 49)}^{2}$ out of ${(y^{2} + 49)}^{2}$ .

$\frac{1}{\sqrt{\frac{1^{2} ((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}{{(y^{2} + 49)}^{2} \cdot 1}}} = x$

Step 2.2.2.11.3

Rearrange the fraction $\frac{1^{2} ((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}{{(y^{2} + 49)}^{2} \cdot 1}$ .

$\frac{1}{\sqrt{{(\frac{1}{y^{2} + 49})}^{2} ((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}} = x$

Step 2.2.2.12

Pull terms out from under the radical.

$\frac{1}{\frac{1}{y^{2} + 49} \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}} = x$

Step 2.2.2.13

Combine $\frac{1}{y^{2} + 49}$ and $\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}$ .

$\frac{1}{\frac{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{y^{2} + 49}} = x$

Step 2.2.3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{y^{2} + 49}{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}} = x$

Step 2.2.4

Multiply $\frac{y^{2} + 49}{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}$ by $1$ .

$\frac{y^{2} + 49}{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}} = x$

Step 2.2.5

Multiply $\frac{y^{2} + 49}{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}$ by $\frac{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}$ .

$\frac{y^{2} + 49}{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}} \cdot \frac{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}} = x$

Step 2.2.6

Combine and simplify the denominator.

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Step 2.2.6.1

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}} = x$

Step 2.2.6.2

Raise $\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}$ to the power of $1$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}^{1} \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}} = x$

Step 2.2.6.3

Raise $\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}$ to the power of $1$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}^{1} {\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}^{1}} = x$

Step 2.2.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}^{1 + 1}} = x$

Step 2.2.6.5

Add $1$ and $1$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}^{2}} = x$

Step 2.2.6.6

Rewrite ${\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}^{2}$ as $(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})$ .

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Step 2.2.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}$ as ${((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}^{\frac{1}{2}}$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{{({((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}^{\frac{1}{2}})}^{2}} = x$

Step 2.2.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{{((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}^{\frac{1}{2} \cdot 2}} = x$

Step 2.2.6.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{{((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}^{\frac{2}{2}}} = x$

Step 2.2.6.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.6.4.1

Cancel the common factor.

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{{((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}^{\frac{2}{2}}} = x$

Step 2.2.6.6.4.2

Rewrite the expression.

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{{((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))}^{1}} = x$

Step 2.2.6.6.5

Simplify.

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} = x$

Step 2.2.7

Multiply $\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}$ by $\frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49 - y \sqrt{y^{2} + 49}}$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} \cdot \frac{y^{2} + 49 - y \sqrt{y^{2} + 49}}{y^{2} + 49 - y \sqrt{y^{2} + 49}} = x$

Step 2.2.8

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} (y^{2} + 49 - y \sqrt{y^{2} + 49})}{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} = x$

Step 2.2.9

Move $y^{2} + 49 + y \sqrt{y^{2} + 49}$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} (y^{2} + 49 - y \sqrt{y^{2} + 49})}{(y^{2} + 49 - y \sqrt{y^{2} + 49}) ((y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49}))} = x$

Step 2.2.10

Expand the denominator using the FOIL method.

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} (y^{2} + 49 - y \sqrt{y^{2} + 49})}{(y^{2} + 49 - y \sqrt{y^{2} + 49}) (y^{4} + y^{2} \cdot 49 + y^{3} (- \sqrt{y^{2} + 49}) + 49 y^{2} + 2401 - 49 (y \sqrt{y^{2} + 49}) + y^{3} \sqrt{y^{2} + 49} + y \sqrt{y^{2} + 49} \cdot 49 + y^{2} (- {\sqrt{y^{2} + 49}}^{2}))} = x$

Step 2.2.11

Simplify.

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Step 2.2.11.1

Subtract $y^{4}$ from $y^{4}$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} (y^{2} + 49 - y \sqrt{y^{2} + 49})}{(y^{2} + 49 - y \sqrt{y^{2} + 49}) (49 y^{2} + 49 y^{2} + 2401 + 0 - 49 y^{2})} = x$

Step 2.2.11.2

Add $49 y^{2} + 49 y^{2} + 2401$ and $0$ .

Step 2.2.11.3

Subtract $49 y^{2}$ from $49 y^{2}$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} (y^{2} + 49 - y \sqrt{y^{2} + 49})}{(y^{2} + 49 - y \sqrt{y^{2} + 49}) (49 y^{2} + 2401 + 0)} = x$

Step 2.2.11.4

Add $49 y^{2} + 2401$ and $0$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} (y^{2} + 49 - y \sqrt{y^{2} + 49})}{(y^{2} + 49 - y \sqrt{y^{2} + 49}) (49 y^{2} + 2401)} = x$

Step 2.2.12

Cancel the common factor of $y^{2} + 49 - y \sqrt{y^{2} + 49}$ .

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Step 2.2.12.1

Cancel the common factor.

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})} (y^{2} + 49 - y \sqrt{y^{2} + 49})}{(y^{2} + 49 - y \sqrt{y^{2} + 49}) (49 y^{2} + 2401)} = x$

Step 2.2.12.2

Rewrite the expression.

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{49 y^{2} + 2401} = x$

Step 2.2.13

Factor $49$ out of $49 y^{2} + 2401$ .

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Step 2.2.13.1

Factor $49$ out of $49 y^{2}$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{49 (y^{2}) + 2401} = x$

Step 2.2.13.2

Factor $49$ out of $2401$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{49 y^{2} + 49 \cdot 49} = x$

Step 2.2.13.3

Factor $49$ out of $49 y^{2} + 49 \cdot 49$ .

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{49 (y^{2} + 49)} = x$

Step 2.2.14

Cancel the common factor of $y^{2} + 49$ .

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Step 2.2.14.1

Cancel the common factor.

$\frac{(y^{2} + 49) \sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{49 (y^{2} + 49)} = x$

Step 2.2.14.2

Rewrite the expression.

$\frac{\sqrt{(y^{2} + 49 + y \sqrt{y^{2} + 49}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{49} = x$

Step 2.3

Simplify the left side.

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Step 2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 49}$ as ${(y^{2} + 49)}^{\frac{1}{2}}$ .

$\frac{\sqrt{(y^{2} + 49 + y {(y^{2} + 49)}^{\frac{1}{2}}) (y^{2} + 49 - y \sqrt{y^{2} + 49})}}{49} = x$

Step 2.3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 49}$ as ${(y^{2} + 49)}^{\frac{1}{2}}$ .

$\frac{\sqrt{(y^{2} + 49 + y {(y^{2} + 49)}^{\frac{1}{2}}) (y^{2} + 49 - y {(y^{2} + 49)}^{\frac{1}{2}})}}{49} = x$

Step 2.3.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(y^{2} + 49 + y {(y^{2} + 49)}^{\frac{1}{2}}) (y^{2} + 49 - y {(y^{2} + 49)}^{\frac{1}{2}})}$ as ${((y^{2} + 49 + y {(y^{2} + 49)}^{\frac{1}{2}}) (y^{2} + 49 - y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}$ .

$\frac{{((y^{2} + 49 + y {(y^{2} + 49)}^{\frac{1}{2}}) (y^{2} + 49 - y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} = x$

Step 2.4

Multiply both sides by $49$ .

$\frac{{((y^{2} + 49 + y {(y^{2} + 49)}^{\frac{1}{2}}) (y^{2} + 49 - y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Simplify $\frac{{((y^{2} + 49 + y {(y^{2} + 49)}^{\frac{1}{2}}) (y^{2} + 49 - y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49$ .

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Step 2.5.1.1.1

Simplify the numerator.

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Step 2.5.1.1.1.1

Expand $(y^{2} + 49 + y {(y^{2} + 49)}^{\frac{1}{2}}) (y^{2} + 49 - y {(y^{2} + 49)}^{\frac{1}{2}})$ by multiplying each term in the first expression by each term in the second expression.

$\frac{{(y^{2} y^{2} + y^{2} \cdot 49 + y^{2} (- y {(y^{2} + 49)}^{\frac{1}{2}}) + 49 y^{2} + 49 \cdot 49 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}}) + y {(y^{2} + 49)}^{\frac{1}{2}} y^{2} + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.2

Combine the opposite terms in $y^{2} y^{2} + y^{2} \cdot 49 + y^{2} (- y {(y^{2} + 49)}^{\frac{1}{2}}) + 49 y^{2} + 49 \cdot 49 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}}) + y {(y^{2} + 49)}^{\frac{1}{2}} y^{2} + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}})$ .

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Step 2.5.1.1.1.2.1

Reorder the factors in the terms $y^{2} (- y {(y^{2} + 49)}^{\frac{1}{2}})$ and $y {(y^{2} + 49)}^{\frac{1}{2}} y^{2}$ .

$\frac{{(y^{2} y^{2} + y^{2} \cdot 49 - y^{2} y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y^{2} + 49 \cdot 49 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}}) + y^{2} y {(y^{2} + 49)}^{\frac{1}{2}} + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.2.2

Add $- y^{2} y {(y^{2} + 49)}^{\frac{1}{2}}$ and $y^{2} y {(y^{2} + 49)}^{\frac{1}{2}}$ .

$\frac{{(y^{2} y^{2} + y^{2} \cdot 49 + 49 y^{2} + 49 \cdot 49 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}}) + 0 + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.2.3

Add $y^{2} y^{2} + y^{2} \cdot 49 + 49 y^{2} + 49 \cdot 49 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}})$ and $0$ .

$\frac{{(y^{2} y^{2} + y^{2} \cdot 49 + 49 y^{2} + 49 \cdot 49 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}}) + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3

Simplify each term.

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Step 2.5.1.1.1.3.1

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 2.5.1.1.1.3.1.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(y^{2 + 2} + y^{2} \cdot 49 + 49 y^{2} + 49 \cdot 49 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}}) + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.1.2

Add $2$ and $2$ .

$\frac{{(y^{4} + y^{2} \cdot 49 + 49 y^{2} + 49 \cdot 49 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}}) + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.2

Move $49$ to the left of $y^{2}$ .

$\frac{{(y^{4} + 49 \cdot y^{2} + 49 y^{2} + 49 \cdot 49 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}}) + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.3

Multiply $49$ by $49$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 + 49 (- y {(y^{2} + 49)}^{\frac{1}{2}}) + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.4

Multiply $- 1$ by $49$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 (y {(y^{2} + 49)}^{\frac{1}{2}}) + y {(y^{2} + 49)}^{\frac{1}{2}} \cdot 49 + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.5

Move $49$ to the left of $y {(y^{2} + 49)}^{\frac{1}{2}}$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 \cdot (y {(y^{2} + 49)}^{\frac{1}{2}}) + y {(y^{2} + 49)}^{\frac{1}{2}} (- y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.6

Rewrite using the commutative property of multiplication.

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y {(y^{2} + 49)}^{\frac{1}{2}} (y {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.7

Multiply $y$ by $y$ by adding the exponents.

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Step 2.5.1.1.1.3.7.1

Move $y$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - (y \cdot y) {(y^{2} + 49)}^{\frac{1}{2}} {(y^{2} + 49)}^{\frac{1}{2}})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.7.2

Multiply $y$ by $y$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{2} {(y^{2} + 49)}^{\frac{1}{2}} {(y^{2} + 49)}^{\frac{1}{2}})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.8

Multiply ${(y^{2} + 49)}^{\frac{1}{2}}$ by ${(y^{2} + 49)}^{\frac{1}{2}}$ by adding the exponents.

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Step 2.5.1.1.1.3.8.1

Move ${(y^{2} + 49)}^{\frac{1}{2}}$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{2} ({(y^{2} + 49)}^{\frac{1}{2}} {(y^{2} + 49)}^{\frac{1}{2}}))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.8.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{2} {(y^{2} + 49)}^{\frac{1}{2} + \frac{1}{2}})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.8.3

Combine the numerators over the common denominator.

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{2} {(y^{2} + 49)}^{\frac{1 + 1}{2}})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.8.4

Add $1$ and $1$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{2} {(y^{2} + 49)}^{\frac{2}{2}})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.8.5

Divide $2$ by $2$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{2} {(y^{2} + 49)}^{1})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.9

Simplify $- y^{2} {(y^{2} + 49)}^{1}$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{2} (y^{2} + 49))}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.10

Apply the distributive property.

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{2} y^{2} - y^{2} \cdot 49)}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.11

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 2.5.1.1.1.3.11.1

Move $y^{2}$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - (y^{2} y^{2}) - y^{2} \cdot 49)}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.11.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{2 + 2} - y^{2} \cdot 49)}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.11.3

Add $2$ and $2$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{4} - y^{2} \cdot 49)}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.3.12

Multiply $49$ by $- 1$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{4} - 49 y^{2})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.4

Combine the opposite terms in $y^{4} + 49 y^{2} + 49 y^{2} + 2401 - 49 y {(y^{2} + 49)}^{\frac{1}{2}} + 49 y {(y^{2} + 49)}^{\frac{1}{2}} - y^{4} - 49 y^{2}$ .

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Step 2.5.1.1.1.4.1

Add $- 49 y {(y^{2} + 49)}^{\frac{1}{2}}$ and $49 y {(y^{2} + 49)}^{\frac{1}{2}}$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 + 0 - y^{4} - 49 y^{2})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.4.2

Add $y^{4} + 49 y^{2} + 49 y^{2} + 2401$ and $0$ .

$\frac{{(y^{4} + 49 y^{2} + 49 y^{2} + 2401 - y^{4} - 49 y^{2})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.4.3

Subtract $y^{4}$ from $y^{4}$ .

$\frac{{(49 y^{2} + 49 y^{2} + 2401 + 0 - 49 y^{2})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.4.4

Add $49 y^{2} + 49 y^{2} + 2401$ and $0$ .

$\frac{{(49 y^{2} + 49 y^{2} + 2401 - 49 y^{2})}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.4.5

Subtract $49 y^{2}$ from $49 y^{2}$ .

$\frac{{(49 y^{2} + 2401 + 0)}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.1.4.6

Add $49 y^{2} + 2401$ and $0$ .

$\frac{{(49 y^{2} + 2401)}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.2

Cancel the common factor of $49$ .

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Step 2.5.1.1.2.1

Cancel the common factor.

$\frac{{(49 y^{2} + 2401)}^{\frac{1}{2}}}{49} \cdot 49 = x \cdot 49$

Step 2.5.1.1.2.2

Rewrite the expression.

${(49 y^{2} + 2401)}^{\frac{1}{2}} = x \cdot 49$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Move $49$ to the left of $x$ .

${(49 y^{2} + 2401)}^{\frac{1}{2}} = 49 x$

Step 2.6

Solve for $y$ .

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Step 2.6.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(49 y^{2} + 2401)}^{\frac{1}{2}})}^{2} = {(49 x)}^{2}$

Step 2.6.2

Simplify the exponent.

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Step 2.6.2.1

Simplify the left side.

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Step 2.6.2.1.1

Simplify ${({(49 y^{2} + 2401)}^{\frac{1}{2}})}^{2}$ .

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Step 2.6.2.1.1.1

Multiply the exponents in ${({(49 y^{2} + 2401)}^{\frac{1}{2}})}^{2}$ .

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Step 2.6.2.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(49 y^{2} + 2401)}^{\frac{1}{2} \cdot 2} = {(49 x)}^{2}$

Step 2.6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.6.2.1.1.1.2.1

Cancel the common factor.

${(49 y^{2} + 2401)}^{\frac{1}{2} \cdot 2} = {(49 x)}^{2}$

Step 2.6.2.1.1.1.2.2

Rewrite the expression.

${(49 y^{2} + 2401)}^{1} = {(49 x)}^{2}$

Step 2.6.2.1.1.2

Simplify.

$49 y^{2} + 2401 = {(49 x)}^{2}$

Step 2.6.2.2

Simplify the right side.

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Step 2.6.2.2.1

Simplify ${(49 x)}^{2}$ .

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Step 2.6.2.2.1.1

Apply the product rule to $49 x$ .

$49 y^{2} + 2401 = 49^{2} x^{2}$

Step 2.6.2.2.1.2

Raise $49$ to the power of $2$ .

$49 y^{2} + 2401 = 2401 x^{2}$

Step 2.6.3

Solve for $y$ .

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Step 2.6.3.1

Subtract $2401$ from both sides of the equation.

$49 y^{2} = 2401 x^{2} - 2401$

Step 2.6.3.2

Divide each term in $49 y^{2} = 2401 x^{2} - 2401$ by $49$ and simplify.

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Step 2.6.3.2.1

Divide each term in $49 y^{2} = 2401 x^{2} - 2401$ by $49$ .

$\frac{49 y^{2}}{49} = \frac{2401 x^{2}}{49} + \frac{- 2401}{49}$

Step 2.6.3.2.2

Simplify the left side.

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Step 2.6.3.2.2.1

Cancel the common factor of $49$ .

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Step 2.6.3.2.2.1.1

Cancel the common factor.

$\frac{49 y^{2}}{49} = \frac{2401 x^{2}}{49} + \frac{- 2401}{49}$

Step 2.6.3.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{2401 x^{2}}{49} + \frac{- 2401}{49}$

Step 2.6.3.2.3

Simplify the right side.

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Step 2.6.3.2.3.1

Simplify each term.

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Step 2.6.3.2.3.1.1

Cancel the common factor of $2401$ and $49$ .

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Step 2.6.3.2.3.1.1.1

Factor $49$ out of $2401 x^{2}$ .

$y^{2} = \frac{49 (49 x^{2})}{49} + \frac{- 2401}{49}$

Step 2.6.3.2.3.1.1.2

Cancel the common factors.

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Step 2.6.3.2.3.1.1.2.1

Factor $49$ out of $49$ .

$y^{2} = \frac{49 (49 x^{2})}{49 (1)} + \frac{- 2401}{49}$

Step 2.6.3.2.3.1.1.2.2

Cancel the common factor.

$y^{2} = \frac{49 (49 x^{2})}{49 \cdot 1} + \frac{- 2401}{49}$

Step 2.6.3.2.3.1.1.2.3

Rewrite the expression.

$y^{2} = \frac{49 x^{2}}{1} + \frac{- 2401}{49}$

Step 2.6.3.2.3.1.1.2.4

Divide $49 x^{2}$ by $1$ .

$y^{2} = 49 x^{2} + \frac{- 2401}{49}$

Step 2.6.3.2.3.1.2

Divide $- 2401$ by $49$ .

$y^{2} = 49 x^{2} - 49$

Step 2.6.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{49 x^{2} - 49}$

Step 2.6.3.4

Simplify $\pm \sqrt{49 x^{2} - 49}$ .

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Step 2.6.3.4.1

Factor $49$ out of $49 x^{2} - 49$ .

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Step 2.6.3.4.1.1

Factor $49$ out of $49 x^{2}$ .

$y = \pm \sqrt{49 (x^{2}) - 49}$

Step 2.6.3.4.1.2

Factor $49$ out of $- 49$ .

$y = \pm \sqrt{49 (x^{2}) + 49 (- 1)}$

Step 2.6.3.4.1.3

Factor $49$ out of $49 (x^{2}) + 49 (- 1)$ .

$y = \pm \sqrt{49 (x^{2} - 1)}$

Step 2.6.3.4.2

Rewrite $1$ as $1^{2}$ .

$y = \pm \sqrt{49 (x^{2} - 1^{2})}$

Step 2.6.3.4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$y = \pm \sqrt{49 (x + 1) (x - 1)}$

Step 2.6.3.4.4

Rewrite $49 (x + 1) (x - 1)$ as $7^{2} ((x + 1^{2}) (x - 1))$ .

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Step 2.6.3.4.4.1

Rewrite $49$ as $7^{2}$ .

$y = \pm \sqrt{7^{2} (x + 1) (x - 1)}$

Step 2.6.3.4.4.2

Rewrite $1$ as $1^{2}$ .

$y = \pm \sqrt{7^{2} (x + 1^{2}) (x - 1)}$

Step 2.6.3.4.4.3

Add parentheses.

$y = \pm \sqrt{7^{2} ((x + 1^{2}) (x - 1))}$

Step 2.6.3.4.5

Pull terms out from under the radical.

$y = \pm 7 \sqrt{(x + 1^{2}) (x - 1)}$

Step 2.6.3.4.6

One to any power is one.

$y = \pm 7 \sqrt{(x + 1) (x - 1)}$

Step 2.6.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.6.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$y = 7 \sqrt{(x + 1) (x - 1)}$

Step 2.6.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 7 \sqrt{(x + 1) (x - 1)}$

Step 2.6.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = 7 \sqrt{(x + 1) (x - 1)}$

$y = - 7 \sqrt{(x + 1) (x - 1)}$

$y = 7 \sqrt{(x + 1) (x - 1)}$

$y = - 7 \sqrt{(x + 1) (x - 1)}$

$y = 7 \sqrt{(x + 1) (x - 1)}$

$y = - 7 \sqrt{(x + 1) (x - 1)}$

$y = 7 \sqrt{(x + 1) (x - 1)}$

$y = - 7 \sqrt{(x + 1) (x - 1)}$

$y = 7 \sqrt{(x + 1) (x - 1)}$

$y = - 7 \sqrt{(x + 1) (x - 1)}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = 7 \sqrt{(x + 1) (x - 1)}, - 7 \sqrt{(x + 1) (x - 1)}$

Step 4

Verify if $f^{- 1} (x) = 7 \sqrt{(x + 1) (x - 1)}, - 7 \sqrt{(x + 1) (x - 1)}$ is the inverse of $f (x) = \sec (\arcsin (\frac{x}{\sqrt{x^{2} + 49}}))$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = \sec (\arcsin (\frac{x}{\sqrt{x^{2} + 49}}))$ and $f^{- 1} (x) = 7 \sqrt{(x + 1) (x - 1)}, - 7 \sqrt{(x + 1) (x - 1)}$ and compare them.

Step 4.2

Find the domain of $7 \sqrt{(x + 1) (x - 1)}$ .

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Step 4.2.1

Set the radicand in $\sqrt{(x + 1) (x - 1)}$ greater than or equal to $0$ to find where the expression is defined.

$(x + 1) (x - 1) \geq 0$

Step 4.2.2

Solve for $x$ .

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Step 4.2.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x - 1 = 0$

Step 4.2.2.2

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.2.2.2.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.2.2.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.2.2.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.2.2.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.2.2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.2.2.4

The final solution is all the values that make $(x + 1) (x - 1) \geq 0$ true.

$x = - 1, 1$

Step 4.2.2.5

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 1$

$x > 1$

Step 4.2.2.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 4.2.2.6.1

Test a value on the interval $x < - 1$ to see if it makes the inequality true.

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Step 4.2.2.6.1.1

Choose a value on the interval $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 4.2.2.6.1.2

Replace $x$ with $- 4$ in the original inequality.

$((- 4) + 1) ((- 4) - 1) \geq 0$

Step 4.2.2.6.1.3

The left side $15$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 4.2.2.6.2

Test a value on the interval $- 1 < x < 1$ to see if it makes the inequality true.

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Step 4.2.2.6.2.1

Choose a value on the interval $- 1 < x < 1$ and see if this value makes the original inequality true.

$x = 0$

Step 4.2.2.6.2.2

Replace $x$ with $0$ in the original inequality.

$((0) + 1) ((0) - 1) \geq 0$

Step 4.2.2.6.2.3

The left side $- 1$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.2.2.6.3

Test a value on the interval $x > 1$ to see if it makes the inequality true.

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Step 4.2.2.6.3.1

Choose a value on the interval $x > 1$ and see if this value makes the original inequality true.

$x = 4$

Step 4.2.2.6.3.2

Replace $x$ with $4$ in the original inequality.

$((4) + 1) ((4) - 1) \geq 0$

Step 4.2.2.6.3.3

The left side $15$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 4.2.2.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 1$ True

$- 1 < x < 1$ False

$x > 1$ True

$x < - 1$ True

$- 1 < x < 1$ False

$x > 1$ True

Step 4.2.2.7

The solution consists of all of the true intervals.

$x \leq - 1$ or $x \geq 1$

Step 4.2.3

The domain is all values of $x$ that make the expression defined.

$(- \infty, - 1] \cup [1, \infty)$

Step 4.3

Since the domain of $f^{- 1} (x) = 7 \sqrt{(x + 1) (x - 1)}, - 7 \sqrt{(x + 1) (x - 1)}$ is not equal to the range of $f (x) = \sec (\arcsin (\frac{x}{\sqrt{x^{2} + 49}}))$ , then $f^{- 1} (x) = 7 \sqrt{(x + 1) (x - 1)}, - 7 \sqrt{(x + 1) (x - 1)}$ is not an inverse of $f (x) = \sec (\arcsin (\frac{x}{\sqrt{x^{2} + 49}}))$ .

There is no inverse

Step 5