Precalculus Examples

${(x - 2)}^{\frac{2}{3}} = 25$

Step 1

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

${({(x - 2)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 25^{\frac{3}{2}}$

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 ${({(x - 2)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

${(x - 2)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 25^{\frac{3}{2}}$

Step 2.1.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(x - 2)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 25^{\frac{3}{2}}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(x - 2)}^{\frac{1}{3} \cdot 3} = \pm 25^{\frac{3}{2}}$

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.

${(x - 2)}^{\frac{1}{3} \cdot 3} = \pm 25^{\frac{3}{2}}$

Step 2.1.1.1.3.2

Rewrite the expression.

${(x - 2)}^{1} = \pm 25^{\frac{3}{2}}$

Step 2.1.1.2

Simplify.

$x - 2 = \pm 25^{\frac{3}{2}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\pm 25^{\frac{3}{2}}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Rewrite $25$ as $5^{2}$ .

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1

Cancel the common factor.

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

Step 2.2.1.2.2

Rewrite the expression.

$x - 2 = \pm 5^{3}$

Step 2.2.1.3

Raise $5$ to the power of $3$ .

$x - 2 = \pm 125$

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.

$x - 2 = 125$

Step 3.2

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

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Step 3.2.1

Add $2$ to both sides of the equation.

$x = 125 + 2$

Step 3.2.2

Add $125$ and $2$ .

$x = 127$

Step 3.3

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

$x - 2 = - 125$

Step 3.4

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

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Step 3.4.1

Add $2$ to both sides of the equation.

$x = - 125 + 2$

Step 3.4.2

Add $- 125$ and $2$ .

$x = - 123$

Step 3.5

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

$x = 127, - 123$