Algebra Examples

${(\frac{a^{5} b^{- 7}}{a^{3} b^{- 9}})}^{- 1} {(\frac{a}{2 b})}^{- 2}$

Step 1

Move $b^{- 9}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{a^{5} b^{- 7} b^{9}}{a^{3}})}^{- 1} {(\frac{a}{2 b})}^{- 2}$

Step 2

Multiply $b^{- 7}$ by $b^{9}$ by adding the exponents.

Tap for more steps...

Step 2.1

Move $b^{9}$ .

${(\frac{a^{5} (b^{9} b^{- 7})}{a^{3}})}^{- 1} {(\frac{a}{2 b})}^{- 2}$

Step 2.2

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

${(\frac{a^{5} b^{9 - 7}}{a^{3}})}^{- 1} {(\frac{a}{2 b})}^{- 2}$

Step 2.3

Subtract $7$ from $9$ .

${(\frac{a^{5} b^{2}}{a^{3}})}^{- 1} {(\frac{a}{2 b})}^{- 2}$

Step 3

Cancel the common factor of $a^{5}$ and $a^{3}$ .

Tap for more steps...

Step 3.1

Factor $a^{3}$ out of $a^{5} b^{2}$ .

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

Step 3.2

Cancel the common factors.

Tap for more steps...

Step 3.2.1

Multiply by $1$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.2.4

Divide $a^{2} b^{2}$ by $1$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.1

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

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

Step 6.2

Apply the product rule to $2 b$ .

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

Step 7

Combine.

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

Step 8

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

Tap for more steps...

Step 8.1

Move $a^{2}$ .

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

Step 8.2

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

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

Step 8.3

Add $2$ and $2$ .

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

Step 9

Multiply $2^{2} b^{2}$ by $1$ .

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

Step 10

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

Tap for more steps...

Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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

Step 11

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

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