Basic Math Examples

${(\frac{a^{3}}{a^{- 2} b})}^{2} \cdot {(\frac{a b^{- 2}}{b^{2}})}^{- 2}$

Step 1

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

${(\frac{a^{3} a^{2}}{b})}^{2} \cdot {(\frac{a b^{- 2}}{b^{2}})}^{- 2}$

Step 2

Multiply $a^{3}$ by $a^{2}$ by adding the exponents.

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Step 2.1

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

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

Step 2.2

Add $3$ and $2$ .

${(\frac{a^{5}}{b})}^{2} \cdot {(\frac{a b^{- 2}}{b^{2}})}^{- 2}$

Step 3

Apply the product rule to $\frac{a^{5}}{b}$ .

$\frac{{(a^{5})}^{2}}{b^{2}} \cdot {(\frac{a b^{- 2}}{b^{2}})}^{- 2}$

Step 4

Multiply the exponents in ${(a^{5})}^{2}$ .

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Step 4.1

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

$\frac{a^{5 \cdot 2}}{b^{2}} \cdot {(\frac{a b^{- 2}}{b^{2}})}^{- 2}$

Step 4.2

Multiply $5$ by $2$ .

$\frac{a^{10}}{b^{2}} \cdot {(\frac{a b^{- 2}}{b^{2}})}^{- 2}$

Step 5

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

$\frac{a^{10}}{b^{2}} \cdot {(\frac{a}{b^{2} b^{2}})}^{- 2}$

Step 6

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

$\frac{a^{10}}{b^{2}} \cdot {(\frac{a}{b^{2 + 2}})}^{- 2}$

Step 6.2

Add $2$ and $2$ .

$\frac{a^{10}}{b^{2}} \cdot {(\frac{a}{b^{4}})}^{- 2}$

Step 7

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

$\frac{a^{10}}{b^{2}} \cdot {(\frac{b^{4}}{a})}^{2}$

Step 8

Apply the product rule to $\frac{b^{4}}{a}$ .

$\frac{a^{10}}{b^{2}} \cdot \frac{{(b^{4})}^{2}}{a^{2}}$

Step 9

Combine.

$\frac{a^{10} {(b^{4})}^{2}}{b^{2} a^{2}}$

Step 10

Cancel the common factor of $a^{10}$ and $a^{2}$ .

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Step 10.1

Factor $a^{2}$ out of $a^{10} {(b^{4})}^{2}$ .

$\frac{a^{2} (a^{8} {(b^{4})}^{2})}{b^{2} a^{2}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $a^{2}$ out of $b^{2} a^{2}$ .

$\frac{a^{2} (a^{8} {(b^{4})}^{2})}{a^{2} b^{2}}$

Step 10.2.2

Cancel the common factor.

$\frac{a^{2} (a^{8} {(b^{4})}^{2})}{a^{2} b^{2}}$

Step 10.2.3

Rewrite the expression.

$\frac{a^{8} {(b^{4})}^{2}}{b^{2}}$

Step 11

Multiply the exponents in ${(b^{4})}^{2}$ .

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Step 11.1

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

$\frac{a^{8} b^{4 \cdot 2}}{b^{2}}$

Step 11.2

Multiply $4$ by $2$ .

$\frac{a^{8} b^{8}}{b^{2}}$

Step 12

Cancel the common factor of $b^{8}$ and $b^{2}$ .

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Step 12.1

Factor $b^{2}$ out of $a^{8} b^{8}$ .

$\frac{b^{2} (a^{8} b^{6})}{b^{2}}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Multiply by $1$ .

$\frac{b^{2} (a^{8} b^{6})}{b^{2} \cdot 1}$

Step 12.2.2

Cancel the common factor.

$\frac{b^{2} (a^{8} b^{6})}{b^{2} \cdot 1}$

Step 12.2.3

Rewrite the expression.

$\frac{a^{8} b^{6}}{1}$

Step 12.2.4

Divide $a^{8} b^{6}$ by $1$ .

$a^{8} b^{6}$