Algebra Examples

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

Step 1

Simplify the numerator.

Tap for more steps...

Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{2} b}$ as ${(a^{2} b)}^{\frac{1}{2}}$ .

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

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a b^{2}}$ as ${(a b^{2})}^{\frac{1}{2}}$ .

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

Step 1.3

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

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

Step 1.4

Multiply the exponents in ${(a^{2})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

Rewrite the expression.

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

Step 1.5

Simplify.

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

Step 1.6

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

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

Step 1.7

Multiply the exponents in ${(b^{2})}^{\frac{1}{2}}$ .

Tap for more steps...

Step 1.7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.7.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.7.2.1

Cancel the common factor.

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

Step 1.7.2.2

Rewrite the expression.

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

Step 1.8

Simplify.

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

Step 1.9

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a b^{\frac{1}{2}} + a^{\frac{1}{2}} b$ .

Tap for more steps...

Step 1.9.1

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a b^{\frac{1}{2}}$ .

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

Step 1.9.2

Factor $a^{\frac{1}{2}} b^{\frac{1}{2}}$ out of $a^{\frac{1}{2}} b$ .

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

Step 1.9.3

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

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

Step 2

Multiply $\frac{a^{\frac{1}{2}} b^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}})}{\sqrt{a b}}$ by $\frac{\sqrt{a b}}{\sqrt{a b}}$ .

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

Step 3

Combine and simplify the denominator.

Tap for more steps...

Step 3.1

Multiply $\frac{a^{\frac{1}{2}} b^{\frac{1}{2}} (a^{\frac{1}{2}} + b^{\frac{1}{2}})}{\sqrt{a b}}$ by $\frac{\sqrt{a b}}{\sqrt{a b}}$ .

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

Step 3.2

Raise $\sqrt{a b}$ to the power of $1$ .

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

Step 3.3

Raise $\sqrt{a b}$ to the power of $1$ .

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Rewrite ${\sqrt{a b}}^{2}$ as $a b$ .

Tap for more steps...

Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a b}$ as ${(a b)}^{\frac{1}{2}}$ .

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

Step 3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Simplify.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Move $a^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 4.2

Move $b^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

Multiply $a$ by $a^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 5.1.1

Move $a^{- \frac{1}{2}}$ .

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

Step 5.1.2

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

Tap for more steps...

Step 5.1.2.1

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

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

Step 5.1.2.2

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

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

Step 5.1.3

Write $1$ as a fraction with a common denominator.

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

Step 5.1.4

Combine the numerators over the common denominator.

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

Step 5.1.5

Add $- 1$ and $2$ .

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

Step 5.2

Multiply $b$ by $b^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 5.2.1

Move $b^{- \frac{1}{2}}$ .

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

Step 5.2.2

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

Tap for more steps...

Step 5.2.2.1

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

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

Step 5.2.2.2

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

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

Step 5.2.3

Write $1$ as a fraction with a common denominator.

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

Step 5.2.4

Combine the numerators over the common denominator.

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

Step 5.2.5

Add $- 1$ and $2$ .

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

Step 6

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

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