Precalculus Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

Step 1.2

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 $b = x$ .

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

Step 2

Multiply $\frac{x}{\sqrt{(1 + x) (1 - x)}}$ by $\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$ .

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

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{x}{\sqrt{(1 + x) (1 - x)}}$ by $\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$ .

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

Step 3.2

Raise $\sqrt{(1 + x) (1 - x)}$ to the power of $1$ .

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

Step 3.3

Raise $\sqrt{(1 + x) (1 - x)}$ to the power of $1$ .

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Rewrite ${\sqrt{(1 + x) (1 - x)}}^{2}$ as $(1 + x) (1 - x)$ .

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $\frac{x \sqrt{(1 + x) (1 - x)}}{(1 + x) (1 - x)}$ .

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

Step 4.2

Apply the product rule to $x \sqrt{(1 + x) (1 - x)}$ .

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

Step 4.3

Apply the product rule to $(1 + x) (1 - x)$ .

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

Step 5

Simplify the numerator.

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Step 5.1

Rewrite ${\sqrt{(1 + x) (1 - x)}}^{2}$ as $(1 + x) (1 - x)$ .

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

Cancel the common factor of $2$ .

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Step 5.1.4.1

Cancel the common factor.

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

Step 5.1.4.2

Rewrite the expression.

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

Step 5.1.5

Simplify.

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

Step 5.2

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Apply the distributive property.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $1$ by $1$ .

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

Step 5.3.1.2

Multiply $- x$ by $1$ .

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

Step 5.3.1.3

Multiply $x$ by $1$ .

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

Step 5.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 5.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1.5.1

Move $x$ .

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

Step 5.3.1.5.2

Multiply $x$ by $x$ .

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

Step 5.3.2

Add $- x$ and $x$ .

$\sqrt{1 + \frac{x^{2} (1 + 0 - x^{2})}{{(1 + x)}^{2} {(1 - x)}^{2}}}$

Step 5.3.3

Add $1$ and $0$ .

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

Step 5.4

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

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

Step 5.5

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 $b = x$ .

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $1 + x$ and ${(1 + x)}^{2}$ .

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Step 6.1.1

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

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

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

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

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

Step 6.1.2.2

Cancel the common factor.

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

Step 6.1.2.3

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $1 - x$ and ${(1 - x)}^{2}$ .

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

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

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

Step 6.2.2.2

Cancel the common factor.

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

Step 6.2.2.3

Rewrite the expression.

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

Step 6.3

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

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 7.1.1

Apply the distributive property.

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

Step 7.1.2

Apply the distributive property.

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

Step 7.1.3

Apply the distributive property.

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

Step 7.2

Simplify and combine like terms.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Multiply $1$ by $1$ .

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

Step 7.2.1.2

Multiply $- x$ by $1$ .

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

Step 7.2.1.3

Multiply $x$ by $1$ .

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

Step 7.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 7.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 7.2.1.5.1

Move $x$ .

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

Step 7.2.1.5.2

Multiply $x$ by $x$ .

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

Step 7.2.2

Add $- x$ and $x$ .

$\sqrt{\frac{1 + 0 - x^{2} + x^{2}}{(1 + x) (1 - x)}}$

Step 7.2.3

Add $1$ and $0$ .

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

Step 7.3

Add $- x^{2}$ and $x^{2}$ .

$\sqrt{\frac{1 + 0}{(1 + x) (1 - x)}}$

Step 7.4

Add $1$ and $0$ .

$\sqrt{\frac{1}{(1 + x) (1 - x)}}$

Step 8

Rewrite $\sqrt{\frac{1}{(1 + x) (1 - x)}}$ as $\frac{\sqrt{1}}{\sqrt{(1 + x) (1 - x)}}$ .

$\frac{\sqrt{1}}{\sqrt{(1 + x) (1 - x)}}$

Step 9

Any root of $1$ is $1$ .

$\frac{1}{\sqrt{(1 + x) (1 - x)}}$

Step 10

Multiply $\frac{1}{\sqrt{(1 + x) (1 - x)}}$ by $\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$ .

$\frac{1}{\sqrt{(1 + x) (1 - x)}} \cdot \frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{1}{\sqrt{(1 + x) (1 - x)}}$ by $\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)}}$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{\sqrt{(1 + x) (1 - x)} \sqrt{(1 + x) (1 - x)}}$

Step 11.2

Raise $\sqrt{(1 + x) (1 - x)}$ to the power of $1$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{1} \sqrt{(1 + x) (1 - x)}}$

Step 11.3

Raise $\sqrt{(1 + x) (1 - x)}$ to the power of $1$ .

$\frac{\sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{1} {\sqrt{(1 + x) (1 - x)}}^{1}}$

Step 11.4

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

$\frac{\sqrt{(1 + x) (1 - x)}}{{\sqrt{(1 + x) (1 - x)}}^{1 + 1}}$

Step 11.5

Add $1$ and $1$ .

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

Step 11.6

Rewrite ${\sqrt{(1 + x) (1 - x)}}^{2}$ as $(1 + x) (1 - x)$ .

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Step 11.6.1

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

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

Step 11.6.2

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

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

Step 11.6.3

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

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

Step 11.6.4

Cancel the common factor of $2$ .

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Step 11.6.4.1

Cancel the common factor.

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

Step 11.6.4.2

Rewrite the expression.

$\frac{\sqrt{(1 + x) (1 - x)}}{{((1 + x) (1 - x))}^{1}}$

Step 11.6.5

Simplify.

$\frac{\sqrt{(1 + x) (1 - x)}}{(1 + x) (1 - x)}$