Basic Math Examples

$\frac{8 a^{- 4} b^{3}}{4 a^{- 2} + 4 b^{2}}$

Step 1

Reduce the expression by cancelling the common factors.

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Step 1.1

Move $a^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{8 b^{3}}{(4 a^{- 2} + 4 b^{2}) a^{4}}$

Step 1.2

Cancel the common factor of $8$ and $4 a^{- 2} + 4 b^{2}$ .

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Step 1.2.1

Factor $4$ out of $8 b^{3}$ .

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

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

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

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 3

Combine $\frac{1 + b^{2} a^{2}}{a^{2}}$ and $a^{4}$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Reduce the expression $\frac{(1 + b^{2} a^{2}) a^{4}}{a^{2}}$ by cancelling the common factors.

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Step 4.1.1

Factor $a^{2}$ out of $(1 + b^{2} a^{2}) a^{4}$ .

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

Step 4.1.2

Multiply by $1$ .

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

Divide $(1 + b^{2} a^{2}) a^{2}$ by $1$ .

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

Step 5

Reorder factors in $\frac{2 b^{3}}{(1 + b^{2} a^{2}) a^{2}}$ .

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