Precalculus Examples

${(3^{- 2} + 4^{- 2})}^{- \frac{1}{2}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{3^{2}} + 4^{- 2})}^{- \frac{1}{2}}$

Step 1.2

Raise $3$ to the power of $2$ .

${(\frac{1}{9} + 4^{- 2})}^{- \frac{1}{2}}$

Step 1.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{9} + \frac{1}{4^{2}})}^{- \frac{1}{2}}$

Step 1.4

Raise $4$ to the power of $2$ .

${(\frac{1}{9} + \frac{1}{16})}^{- \frac{1}{2}}$

Step 2

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

${(\frac{1}{9} \cdot \frac{16}{16} + \frac{1}{16})}^{- \frac{1}{2}}$

Step 3

To write $\frac{1}{16}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

${(\frac{1}{9} \cdot \frac{16}{16} + \frac{1}{16} \cdot \frac{9}{9})}^{- \frac{1}{2}}$

Step 4

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{9}$ by $\frac{16}{16}$ .

${(\frac{16}{9 \cdot 16} + \frac{1}{16} \cdot \frac{9}{9})}^{- \frac{1}{2}}$

Step 4.2

Multiply $9$ by $16$ .

${(\frac{16}{144} + \frac{1}{16} \cdot \frac{9}{9})}^{- \frac{1}{2}}$

Step 4.3

Multiply $\frac{1}{16}$ by $\frac{9}{9}$ .

${(\frac{16}{144} + \frac{9}{16 \cdot 9})}^{- \frac{1}{2}}$

Step 4.4

Multiply $16$ by $9$ .

${(\frac{16}{144} + \frac{9}{144})}^{- \frac{1}{2}}$

Step 5

Combine the numerators over the common denominator.

${(\frac{16 + 9}{144})}^{- \frac{1}{2}}$

Step 6

Add $16$ and $9$ .

${(\frac{25}{144})}^{- \frac{1}{2}}$

Step 7

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

${(\frac{144}{25})}^{\frac{1}{2}}$

Step 8

Apply the product rule to $\frac{144}{25}$ .

$\frac{144^{\frac{1}{2}}}{25^{\frac{1}{2}}}$

Step 9

Simplify the numerator.

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Step 9.1

Rewrite $144$ as $12^{2}$ .

$\frac{{(12^{2})}^{\frac{1}{2}}}{25^{\frac{1}{2}}}$

Step 9.2

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

$\frac{12^{2 (\frac{1}{2})}}{25^{\frac{1}{2}}}$

Step 9.3

Cancel the common factor of $2$ .

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Step 9.3.1

Cancel the common factor.

$\frac{12^{2 (\frac{1}{2})}}{25^{\frac{1}{2}}}$

Step 9.3.2

Rewrite the expression.

$\frac{12^{1}}{25^{\frac{1}{2}}}$

Step 9.4

Evaluate the exponent.

$\frac{12}{25^{\frac{1}{2}}}$

Step 10

Simplify the denominator.

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Step 10.1

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

$\frac{12}{{(5^{2})}^{\frac{1}{2}}}$

Step 10.2

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

$\frac{12}{5^{2 (\frac{1}{2})}}$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Cancel the common factor.

$\frac{12}{5^{2 (\frac{1}{2})}}$

Step 10.3.2

Rewrite the expression.

$\frac{12}{5^{1}}$

Step 10.4

Evaluate the exponent.

$\frac{12}{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{12}{5}$

Decimal Form:

$2.4$

Mixed Number Form:

$2 \frac{2}{5}$