Basic Math Examples

\frac{\frac{\sqrt{b}}{3 a b}}{\frac{1}{a^{2}} + \frac{1}{a b^{2}}}

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

\frac{\sqrt{b}}{3 a b} \cdot \frac{1}{\frac{1}{a^{2}} + \frac{1}{a b^{2}}}

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

Step 2

Simplify the denominator.

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Step 2.1

To write

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

$\frac{1}{a^{2}}$ as a fraction with a common denominator, multiply by

\frac{b^{2}}{b^{2}}

$\frac{b^{2}}{b^{2}}$ .

\frac{\sqrt{b}}{3 a b} \cdot \frac{1}{\frac{1}{a^{2}} \cdot \frac{b^{2}}{b^{2}} + \frac{1}{a b^{2}}}

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

Step 2.2

To write

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

$\frac{1}{a b^{2}}$ as a fraction with a common denominator, multiply by

\frac{a}{a}

$\frac{a}{a}$ .

\frac{\sqrt{b}}{3 a b} \cdot \frac{1}{\frac{1}{a^{2}} \cdot \frac{b^{2}}{b^{2}} + \frac{1}{a b^{2}} \cdot \frac{a}{a}}

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

Step 2.3

Write each expression with a common denominator of

a^{2} b^{2}

$a^{2} b^{2}$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 2.3.1

Multiply

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

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

\frac{b^{2}}{b^{2}}

$\frac{b^{2}}{b^{2}}$ .

\frac{\sqrt{b}}{3 a b} \cdot \frac{1}{\frac{b^{2}}{a^{2} b^{2}} + \frac{1}{a b^{2}} \cdot \frac{a}{a}}

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

Step 2.3.2

Multiply

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

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

\frac{a}{a}

$\frac{a}{a}$ .

\frac{\sqrt{b}}{3 a b} \cdot \frac{1}{\frac{b^{2}}{a^{2} b^{2}} + \frac{a}{a b^{2} a}}

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

Step 2.3.3

Raise

a

$a$ to the power of

1

$1$ .

\frac{\sqrt{b}}{3 a b} \cdot \frac{1}{\frac{b^{2}}{a^{2} b^{2}} + \frac{a}{a^{1} a b^{2}}}

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

Step 2.3.4

Raise

a

$a$ to the power of

1

$1$ .

\frac{\sqrt{b}}{3 a b} \cdot \frac{1}{\frac{b^{2}}{a^{2} b^{2}} + \frac{a}{a^{1} a^{1} b^{2}}}

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

Step 2.3.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.3.6

Add

$1$ and

$1$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Combine fractions.

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Step 3.1

Combine.

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

Step 3.2

Multiply

$\sqrt{b}$ by

$1$ .

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

Step 4

Simplify the denominator.

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Step 4.1

Combine

$3$ and

$\frac{b^{2} + a}{a^{2} b^{2}}$ .

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

Step 4.2

Combine

$a$ and

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

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

Step 4.3

Combine

$b$ and

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

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

Step 5

Remove unnecessary parentheses.

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

Step 6

Simplify the denominator.

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Step 6.1

Reduce the expression

$\frac{b a \cdot 3 (b^{2} + a)}{a^{2} b^{2}}$ by cancelling the common factors.

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Step 6.1.1

Factor

$b$ out of

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

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

Step 6.1.2

Factor

$b$ out of

$a^{2} b^{2}$ .

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

Step 6.1.3

Cancel the common factor.

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

Step 6.1.4

Rewrite the expression.

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

Step 6.2

Cancel the common factor of

$a$ and

$a^{2}$ .

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Step 6.2.1

Factor

$a$ out of

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

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

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor

$a$ out of

$a^{2} b$ .

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

Step 6.2.2.2

Cancel the common factor.

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

Step 6.2.2.3

Rewrite the expression.

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

Step 7

Multiply the numerator by the reciprocal of the denominator.

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

Step 8

Combine

$\sqrt{b}$ and

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

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