Trigonometry Examples

$f g = \sqrt{32 x - 58 + 10}$

Step 1

Rewrite the equation as $\sqrt{32 x - 58 + 10} = f g$ .

$\sqrt{32 x - 58 + 10} = f g$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{32 x - 58 + 10}}^{2} = {(f g)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{32 x - 58 + 10}$ as ${(32 x - 58 + 10)}^{\frac{1}{2}}$ .

${({(32 x - 58 + 10)}^{\frac{1}{2}})}^{2} = {(f g)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(32 x - 58 + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(32 x - 58 + 10)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(32 x - 58 + 10)}^{\frac{1}{2} \cdot 2} = {(f g)}^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(32 x - 58 + 10)}^{\frac{1}{2} \cdot 2} = {(f g)}^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(32 x - 58 + 10)}^{1} = {(f g)}^{2}$

Step 3.2.1.2

Add $- 58$ and $10$ .

${(32 x - 48)}^{1} = {(f g)}^{2}$

Step 3.2.1.3

Simplify.

$32 x - 48 = {(f g)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the product rule to $f g$ .

$32 x - 48 = f^{2} g^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Add $48$ to both sides of the equation.

$32 x = f^{2} g^{2} + 48$

Step 4.2

Divide each term in $32 x = f^{2} g^{2} + 48$ by $32$ and simplify.

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Step 4.2.1

Divide each term in $32 x = f^{2} g^{2} + 48$ by $32$ .

$\frac{32 x}{32} = \frac{f^{2} g^{2}}{32} + \frac{48}{32}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $32$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{32 x}{32} = \frac{f^{2} g^{2}}{32} + \frac{48}{32}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{f^{2} g^{2}}{32} + \frac{48}{32}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $48$ and $32$ .

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Step 4.2.3.1.1

Factor $16$ out of $48$ .

$x = \frac{f^{2} g^{2}}{32} + \frac{16 (3)}{32}$

Step 4.2.3.1.2

Cancel the common factors.

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Step 4.2.3.1.2.1

Factor $16$ out of $32$ .

$x = \frac{f^{2} g^{2}}{32} + \frac{16 \cdot 3}{16 \cdot 2}$

Step 4.2.3.1.2.2

Cancel the common factor.

$x = \frac{f^{2} g^{2}}{32} + \frac{16 \cdot 3}{16 \cdot 2}$

Step 4.2.3.1.2.3

Rewrite the expression.

$x = \frac{f^{2} g^{2}}{32} + \frac{3}{2}$