Algebra Examples

${({(a^{- 2})}^{- 1})}^{- 1} \div {(a \div a^{- 1})}^{2}$

Step 1

Rewrite the division as a fraction.

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

Step 2

Simplify the expression.

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Step 2.1

Move ${({(a^{- 2})}^{- 1})}^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2

Move ${(a^{- 2})}^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 2.3

Move $a^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

Cancel the common factor of $a$ and $a^{- 1}$ .

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Step 3.1

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

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

Step 3.2

Factor $a$ out of $a^{1}$ .

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

Step 3.3

Cancel the common factors.

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Step 3.3.1

Factor $a$ out of $a^{- 1}$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Simplify the denominator.

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Step 5.1

Apply the product rule to $\frac{1}{\frac{1}{a^{2}}}$ .

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

Step 5.2

One to any power is one.

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

Step 5.3

Simplify the denominator.

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Step 5.3.1

Apply the product rule to $\frac{1}{a^{2}}$ .

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

Step 5.3.2

One to any power is one.

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

Step 5.3.3

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

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Step 5.3.3.1

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

$\frac{1}{\frac{1}{\frac{1}{a^{2 \cdot 2}}} a^{2}}$

Step 5.3.3.2

Multiply $2$ by $2$ .

$\frac{1}{\frac{1}{\frac{1}{a^{4}}} a^{2}}$

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{1 a^{4} a^{2}}$

Step 5.5

Multiply $a^{4}$ by $1$ .

$\frac{1}{a^{4} a^{2}}$

Step 6

Multiply $a^{4}$ by $a^{2}$ by adding the exponents.

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Step 6.1

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

$\frac{1}{a^{4 + 2}}$

Step 6.2

Add $4$ and $2$ .

$\frac{1}{a^{6}}$