Algebra Examples

$\sqrt[3]{{(x - 2)}^{2}} = 5$

Step 1

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

${\sqrt[3]{{(x - 2)}^{2}}}^{3} = 5^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{{(x - 2)}^{2}}$ as ${(x - 2)}^{\frac{2}{3}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

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.

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

Step 2.2.1.2.2

Rewrite the expression.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

${(x - 2)}^{2} = 125$

Step 3

Solve for $x$ .

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Step 3.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 2 = \pm \sqrt{125}$

Step 3.2

Simplify $\pm \sqrt{125}$ .

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Step 3.2.1

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 3.2.1.1

Factor $25$ out of $125$ .

$x - 2 = \pm \sqrt{25 (5)}$

Step 3.2.1.2

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

$x - 2 = \pm \sqrt{5^{2} \cdot 5}$

Step 3.2.2

Pull terms out from under the radical.

$x - 2 = \pm 5 \sqrt{5}$

Step 3.3

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

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Step 3.3.1

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

$x - 2 = 5 \sqrt{5}$

Step 3.3.2

Add $2$ to both sides of the equation.

$x = 5 \sqrt{5} + 2$

Step 3.3.3

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

$x - 2 = - 5 \sqrt{5}$

Step 3.3.4

Add $2$ to both sides of the equation.

$x = - 5 \sqrt{5} + 2$

Step 3.3.5

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

$x = 5 \sqrt{5} + 2, - 5 \sqrt{5} + 2$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 5 \sqrt{5} + 2, - 5 \sqrt{5} + 2$

Decimal Form:

$x = 13.18033988 \dots, - 9.18033988 \dots$