Precalculus Examples

$z^{\frac{2}{3}} - \frac{1}{25} = 0$

Step 1

Add $\frac{1}{25}$ to both sides of the equation.

$z^{\frac{2}{3}} = \frac{1}{25}$

Step 2

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

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

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

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

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Step 3.1.1.1

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

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Step 3.1.1.1.1

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

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

Step 3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.1.1.1.2.1

Cancel the common factor.

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

Step 3.1.1.1.2.2

Rewrite the expression.

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

Step 3.1.1.1.3

Cancel the common factor of $3$ .

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Step 3.1.1.1.3.1

Cancel the common factor.

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

Step 3.1.1.1.3.2

Rewrite the expression.

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

Step 3.1.1.2

Simplify.

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

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\pm {(\frac{1}{25})}^{\frac{3}{2}}$ .

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Step 3.2.1.1

Simplify the expression.

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Step 3.2.1.1.1

Apply the product rule to $\frac{1}{25}$ .

$z = \pm \frac{1^{\frac{3}{2}}}{25^{\frac{3}{2}}}$

Step 3.2.1.1.2

One to any power is one.

$z = \pm \frac{1}{25^{\frac{3}{2}}}$

Step 3.2.1.2

Simplify the denominator.

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Step 3.2.1.2.1

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

$z = \pm \frac{1}{{(5^{2})}^{\frac{3}{2}}}$

Step 3.2.1.2.2

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

$z = \pm \frac{1}{5^{2 (\frac{3}{2})}}$

Step 3.2.1.2.3

Cancel the common factor of $2$ .

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Step 3.2.1.2.3.1

Cancel the common factor.

$z = \pm \frac{1}{5^{2 (\frac{3}{2})}}$

Step 3.2.1.2.3.2

Rewrite the expression.

$z = \pm \frac{1}{5^{3}}$

Step 3.2.1.2.4

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

$z = \pm \frac{1}{125}$

Step 4

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

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Step 4.1

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

$z = \frac{1}{125}$

Step 4.2

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

$z = - \frac{1}{125}$

Step 4.3

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

$z = \frac{1}{125}, - \frac{1}{125}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$z = \frac{1}{125}, - \frac{1}{125}$

Decimal Form:

$z = 0.008, - 0.008$