Calculus Examples

$- 3^{2} {(3^{2} + 16)}^{- \frac{3}{2}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$- 1 \cdot 9 {(3^{2} + 16)}^{- \frac{3}{2}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.2

Multiply $- 1$ by $9$ .

$- 9 {(3^{2} + 16)}^{- \frac{3}{2}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.3

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

$- 9 {(9 + 16)}^{- \frac{3}{2}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.4

Add $9$ and $16$ .

$- 9 \cdot 25^{- \frac{3}{2}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.5

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

$- 9 \cdot \frac{1}{25^{\frac{3}{2}}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.6

Simplify the denominator.

Tap for more steps...

Step 1.6.1

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

$- 9 \cdot \frac{1}{{(5^{2})}^{\frac{3}{2}}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.6.2

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

$- 9 \cdot \frac{1}{5^{2 (\frac{3}{2})}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.6.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.6.3.1

Cancel the common factor.

$- 9 \cdot \frac{1}{5^{2 (\frac{3}{2})}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.6.3.2

Rewrite the expression.

$- 9 \cdot \frac{1}{5^{3}} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.6.4

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

$- 9 \cdot \frac{1}{125} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.7

Combine $- 9$ and $\frac{1}{125}$ .

$\frac{- 9}{125} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.8

Move the negative in front of the fraction.

$- \frac{9}{125} + {(3^{2} + 16)}^{- \frac{1}{2}}$

Step 1.9

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

$- \frac{9}{125} + {(9 + 16)}^{- \frac{1}{2}}$

Step 1.10

Add $9$ and $16$ .

$- \frac{9}{125} + 25^{- \frac{1}{2}}$

Step 1.11

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

$- \frac{9}{125} + \frac{1}{25^{\frac{1}{2}}}$

Step 1.12

Simplify the denominator.

Tap for more steps...

Step 1.12.1

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

$- \frac{9}{125} + \frac{1}{{(5^{2})}^{\frac{1}{2}}}$

Step 1.12.2

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

$- \frac{9}{125} + \frac{1}{5^{2 (\frac{1}{2})}}$

Step 1.12.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.12.3.1

Cancel the common factor.

$- \frac{9}{125} + \frac{1}{5^{2 (\frac{1}{2})}}$

Step 1.12.3.2

Rewrite the expression.

$- \frac{9}{125} + \frac{1}{5^{1}}$

Step 1.12.4

Evaluate the exponent.

$- \frac{9}{125} + \frac{1}{5}$

Step 2

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$- \frac{9}{125} + \frac{1}{5} \cdot \frac{25}{25}$

Step 3

Write each expression with a common denominator of $125$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.1

Multiply $\frac{1}{5}$ by $\frac{25}{25}$ .

$- \frac{9}{125} + \frac{25}{5 \cdot 25}$

Step 3.2

Multiply $5$ by $25$ .

$- \frac{9}{125} + \frac{25}{125}$

Step 4

Combine the numerators over the common denominator.

$\frac{- 9 + 25}{125}$

Step 5

Add $- 9$ and $25$ .

$\frac{16}{125}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{16}{125}$

Decimal Form:

$0.128$