Precalculus Examples

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

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

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}}

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 1.2

Raise

3

$3$ to the power of

2

$2$ .

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

$b^{- n} = \frac{1}{b^{n}}$ .

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

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

Step 1.4

Raise

4

$4$ to the power of

2

$2$ .

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

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

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

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

Step 2

To write

\frac{1}{9}

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

\frac{16}{16}

$\frac{16}{16}$ .

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

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

Step 3

To write

\frac{1}{16}

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

\frac{9}{9}

$\frac{9}{9}$ .

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

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

Step 4

Write each expression with a common denominator of

144

$144$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 4.1

Multiply

\frac{1}{9}

$\frac{1}{9}$ by

\frac{16}{16}

$\frac{16}{16}$ .

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

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

Step 4.2

Multiply

9

$9$ by

16

$16$ .

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

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

Step 4.3

Multiply

\frac{1}{16}

$\frac{1}{16}$ by

\frac{9}{9}

$\frac{9}{9}$ .

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

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

Step 4.4

Multiply

16

$16$ by

9

$9$ .

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

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

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

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

Step 5

Combine the numerators over the common denominator.

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

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

Step 6

Add

16

$16$ and

9

$9$ .

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

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

Step 7

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

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

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

Step 8

Apply the product rule to

\frac{144}{25}

$\frac{144}{25}$ .

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

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

Step 9

Simplify the numerator.

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Step 9.1

Rewrite

144

$144$ as

12^{2}

$12^{2}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 9.3

Cancel the common factor of

2

$2$ .

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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:

$2 \frac{2}{5}$