Precalculus Examples

${(u - 8)}^{\frac{2}{3}} = 5$

Step 1

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${({(u - 8)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 5^{\frac{3}{2}}$

Step 2

Simplify the left side.

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Step 2.1

Simplify ${({(u - 8)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.1.1

Multiply the exponents in ${({(u - 8)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.1.1.1

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

${(u - 8)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 5^{\frac{3}{2}}$

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Cancel the common factor.

${(u - 8)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 5^{\frac{3}{2}}$

Step 2.1.1.2.2

Rewrite the expression.

${(u - 8)}^{\frac{1}{3} \cdot 3} = \pm 5^{\frac{3}{2}}$

Step 2.1.1.3

Cancel the common factor of $3$ .

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Step 2.1.1.3.1

Cancel the common factor.

${(u - 8)}^{\frac{1}{3} \cdot 3} = \pm 5^{\frac{3}{2}}$

Step 2.1.1.3.2

Rewrite the expression.

${(u - 8)}^{1} = \pm 5^{\frac{3}{2}}$

Step 2.1.2

Simplify.

$u - 8 = \pm 5^{\frac{3}{2}}$

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.

$u - 8 = 5^{\frac{3}{2}}$

Step 3.2

Add $8$ to both sides of the equation.

$u = 5^{\frac{3}{2}} + 8$

Step 3.3

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

$u - 8 = - 5^{\frac{3}{2}}$

Step 3.4

Add $8$ to both sides of the equation.

$u = - 5^{\frac{3}{2}} + 8$

Step 3.5

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

$u = 5^{\frac{3}{2}} + 8, - 5^{\frac{3}{2}} + 8$

Step 4

The result can be shown in multiple forms.

Exact Form:

$u = 5^{\frac{3}{2}} + 8, - 5^{\frac{3}{2}} + 8$

Decimal Form:

$u = 19.18033988 \dots, - 3.18033988 \dots$