Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Combine $3$ and $\frac{1}{a}$ .

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

Step 1.3

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

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

Step 1.4

To write $\frac{3}{a}$ as a fraction with a common denominator, multiply by $\frac{b^{3}}{b^{3}}$ .

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

Step 1.5

To write $\frac{1}{b^{3}}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 1.6

Write each expression with a common denominator of $a b^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.6.1

Multiply $\frac{3}{a}$ by $\frac{b^{3}}{b^{3}}$ .

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

Step 1.6.2

Multiply $\frac{1}{b^{3}}$ by $\frac{a}{a}$ .

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

Step 1.6.3

Reorder the factors of $a b^{3}$ .

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

Step 1.7

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $a$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Cancel the common factor.

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

Step 4.4

Rewrite the expression.

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

Step 5

Multiply $\frac{3 b^{3} + a}{b^{3}}$ by $\frac{a}{4 + b^{2} a^{2}}$ .

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

Step 6

Reorder factors in $\frac{(3 b^{3} + a) a}{b^{3} (4 + b^{2} a^{2})}$ .

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