Precalculus Examples

$x^{\frac{4}{3}} = 81$

Step 1

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

${(x^{\frac{4}{3}})}^{\frac{3}{4}} = \pm 81^{\frac{3}{4}}$

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

$x^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 81^{\frac{3}{4}}$

Step 2.1.1.1.2

Cancel the common factor of $4$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

$x^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 81^{\frac{3}{4}}$

Step 2.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 81^{\frac{3}{4}}$

Step 2.1.1.1.3

Cancel the common factor of $3$ .

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Step 2.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 81^{\frac{3}{4}}$

Step 2.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm 81^{\frac{3}{4}}$

Step 2.1.1.2

Simplify.

$x = \pm 81^{\frac{3}{4}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\pm 81^{\frac{3}{4}}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Rewrite $81$ as $3^{4}$ .

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

Cancel the common factor of $4$ .

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Step 2.2.1.2.1

Cancel the common factor.

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

Step 2.2.1.2.2

Rewrite the expression.

$x = \pm 3^{3}$

Step 2.2.1.3

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

$x = \pm 27$

Step 3

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

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Step 3.1

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

$x = 27$

Step 3.2

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

$x = - 27$

Step 3.3

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

$x = 27, - 27$