Basic Math Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $a^{2} b^{2}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Raise $a$ to the power of $1$ .

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

Step 2.3.4

Raise $a$ to the power of $1$ .

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

Step 2.3.5

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

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

Step 6.1

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

Tap for more steps...

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 $a^{2}$ .

Tap for more steps...

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.

Tap for more steps...

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

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