Precalculus Examples

${(2 a^{- 2} + a)}^{- 2}$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Combine $2$ and $\frac{1}{a^{2}}$ .

${(\frac{2}{a^{2}} + a)}^{- 2}$

Step 2

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

$\frac{1}{{(\frac{2}{a^{2}} + a)}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

To write $a$ as a fraction with a common denominator, multiply by $\frac{a^{2}}{a^{2}}$ .

$\frac{1}{{(\frac{2}{a^{2}} + \frac{a \cdot a^{2}}{a^{2}})}^{2}}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{1}{{(\frac{2 + a \cdot a^{2}}{a^{2}})}^{2}}$

Step 3.3

Multiply $a$ by $a^{2}$ by adding the exponents.

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Step 3.3.1

Multiply $a$ by $a^{2}$ .

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Step 3.3.1.1

Raise $a$ to the power of $1$ .

$\frac{1}{{(\frac{2 + a^{1} a^{2}}{a^{2}})}^{2}}$

Step 3.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{(\frac{2 + a^{1 + 2}}{a^{2}})}^{2}}$

Step 3.3.2

Add $1$ and $2$ .

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

Step 3.4

Apply the product rule to $\frac{2 + a^{3}}{a^{2}}$ .

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

Step 3.5

Multiply the exponents in ${(a^{2})}^{2}$ .

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Step 3.5.1

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

$\frac{1}{\frac{{(2 + a^{3})}^{2}}{a^{2 \cdot 2}}}$

Step 3.5.2

Multiply $2$ by $2$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Multiply $\frac{a^{4}}{{(2 + a^{3})}^{2}}$ by $1$ .

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