Trigonometry Examples

${(x - 5)}^{\frac{3}{4}} = 27$

Step 1

Raise each side of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent on the left side.

${({(x - 5)}^{\frac{3}{4}})}^{\frac{4}{3}} = 27^{\frac{4}{3}}$

Step 2

Simplify the exponent.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify ${({(x - 5)}^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(x - 5)}^{\frac{3}{4}})}^{\frac{4}{3}}$ .

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Step 2.1.1.1.1

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

${(x - 5)}^{\frac{3}{4} \cdot \frac{4}{3}} = 27^{\frac{4}{3}}$

Step 2.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(x - 5)}^{\frac{3}{4} \cdot \frac{4}{3}} = 27^{\frac{4}{3}}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(x - 5)}^{\frac{1}{4} \cdot 4} = 27^{\frac{4}{3}}$

Step 2.1.1.1.3

Cancel the common factor of $4$ .

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Step 2.1.1.1.3.1

Cancel the common factor.

${(x - 5)}^{\frac{1}{4} \cdot 4} = 27^{\frac{4}{3}}$

Step 2.1.1.1.3.2

Rewrite the expression.

${(x - 5)}^{1} = 27^{\frac{4}{3}}$

Step 2.1.1.2

Simplify.

$x - 5 = 27^{\frac{4}{3}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $27^{\frac{4}{3}}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Rewrite $27$ as $3^{3}$ .

$x - 5 = {(3^{3})}^{\frac{4}{3}}$

Step 2.2.1.1.2

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

$x - 5 = 3^{3 (\frac{4}{3})}$

Step 2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.2.1

Cancel the common factor.

$x - 5 = 3^{3 (\frac{4}{3})}$

Step 2.2.1.2.2

Rewrite the expression.

$x - 5 = 3^{4}$

Step 2.2.1.3

Raise $3$ to the power of $4$ .

$x - 5 = 81$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Add $5$ to both sides of the equation.

$x = 81 + 5$

Step 3.2

Add $81$ and $5$ .

$x = 86$