Basic Math Examples

${(z^{2} + 4)}^{\frac{2}{3}} = 25$

Step 1

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

${({(z^{2} + 4)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 25^{\frac{3}{2}}$

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 ${({(z^{2} + 4)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(z^{2} + 4)}^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 2.1.1.1.1

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

${(z^{2} + 4)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 25^{\frac{3}{2}}$

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.

${(z^{2} + 4)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 25^{\frac{3}{2}}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(z^{2} + 4)}^{\frac{1}{3} \cdot 3} = \pm 25^{\frac{3}{2}}$

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.

${(z^{2} + 4)}^{\frac{1}{3} \cdot 3} = \pm 25^{\frac{3}{2}}$

Step 2.1.1.1.3.2

Rewrite the expression.

${(z^{2} + 4)}^{1} = \pm 25^{\frac{3}{2}}$

Step 2.1.1.2

Simplify.

$z^{2} + 4 = \pm 25^{\frac{3}{2}}$

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 $5^{2}$ .

$z^{2} + 4 = \pm {(5^{2})}^{\frac{3}{2}}$

Step 2.2.1.1.2

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

$z^{2} + 4 = \pm 5^{2 (\frac{3}{2})}$

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.

$z^{2} + 4 = \pm 5^{2 (\frac{3}{2})}$

Step 2.2.1.2.2

Rewrite the expression.

$z^{2} + 4 = \pm 5^{3}$

Step 2.2.1.3

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

$z^{2} + 4 = \pm 125$

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.

$z^{2} + 4 = 125$

Step 3.2

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

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Step 3.2.1

Subtract $4$ from both sides of the equation.

$z^{2} = 125 - 4$

Step 3.2.2

Subtract $4$ from $125$ .

$z^{2} = 121$

Step 3.3

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

$z = \pm \sqrt{121}$

Step 3.4

Simplify $\pm \sqrt{121}$ .

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Step 3.4.1

Rewrite $121$ as $11^{2}$ .

$z = \pm \sqrt{11^{2}}$

Step 3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$z = \pm 11$

Step 3.5

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

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Step 3.5.1

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

$z = 11$

Step 3.5.2

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

$z = - 11$

Step 3.5.3

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

$z = 11, - 11$

Step 3.6

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

$z^{2} + 4 = - 125$

Step 3.7

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

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Step 3.7.1

Subtract $4$ from both sides of the equation.

$z^{2} = - 125 - 4$

Step 3.7.2

Subtract $4$ from $- 125$ .

$z^{2} = - 129$

Step 3.8

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

$z = \pm \sqrt{- 129}$

Step 3.9

Simplify $\pm \sqrt{- 129}$ .

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Step 3.9.1

Rewrite $- 129$ as $- 1 (129)$ .

$z = \pm \sqrt{- 1 (129)}$

Step 3.9.2

Rewrite $\sqrt{- 1 (129)}$ as $\sqrt{- 1} \cdot \sqrt{129}$ .

$z = \pm \sqrt{- 1} \cdot \sqrt{129}$

Step 3.9.3

Rewrite $\sqrt{- 1}$ as $i$ .

$z = \pm i \sqrt{129}$

Step 3.10

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

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Step 3.10.1

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

$z = i \sqrt{129}$

Step 3.10.2

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

$z = - i \sqrt{129}$

Step 3.10.3

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

$z = i \sqrt{129}, - i \sqrt{129}$

Step 3.11

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

$z = 11, - 11, i \sqrt{129}, - i \sqrt{129}$