Precalculus Examples

${(\frac{9}{25})}^{- \frac{1}{2}}$

Step 1

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{25}{9})}^{\frac{1}{2}}$

Step 2

Apply the product rule to $\frac{25}{9}$ .

$\frac{25^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{{(5^{2})}^{\frac{1}{2}}}{9^{\frac{1}{2}}}$

Step 3.2

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

$\frac{5^{2 (\frac{1}{2})}}{9^{\frac{1}{2}}}$

Step 3.3

Cancel the common factor of $2$ .

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Step 3.3.1

Cancel the common factor.

$\frac{5^{2 (\frac{1}{2})}}{9^{\frac{1}{2}}}$

Step 3.3.2

Rewrite the expression.

$\frac{5^{1}}{9^{\frac{1}{2}}}$

Step 3.4

Evaluate the exponent.

$\frac{5}{9^{\frac{1}{2}}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $9$ as $3^{2}$ .

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

Step 4.2

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

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

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$\frac{5}{3^{1}}$

Step 4.4

Evaluate the exponent.

$\frac{5}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{3}$

Decimal Form:

$1.\overline{6}$

Mixed Number Form:

$1 \frac{2}{3}$