Basic Math Examples

\sqrt{16 a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}

$\sqrt{16 a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite

16 a

$16 a$ as

{(2^{2})}^{2} a

${(2^{2})}^{2} a$ .

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Step 1.1.1

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

\sqrt{4^{2} a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}

$\sqrt{4^{2} a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}$

Step 1.1.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

\sqrt{{(2^{2})}^{2} a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}

$\sqrt{{(2^{2})}^{2} a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}$

\sqrt{{(2^{2})}^{2} a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}

$\sqrt{{(2^{2})}^{2} a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}$

Step 1.2

Pull terms out from under the radical.

2^{2} \sqrt{a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}

$2^{2} \sqrt{a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}$

Step 1.3

Raise

2

$2$ to the power of

2

$2$ .

4 \sqrt{a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}

$4 \sqrt{a} - \sqrt{a b^{4}} \div \sqrt{4 a} + \sqrt{a b^{2}}$

Step 1.4

Rewrite the division as a fraction.

4 \sqrt{a} - \frac{\sqrt{a b^{4}}}{\sqrt{4 a}} + \sqrt{a b^{2}}

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

Step 1.5

Combine

\sqrt{a b^{4}}

$\sqrt{a b^{4}}$ and

\sqrt{4 a}

$\sqrt{4 a}$ into a single radical.

4 \sqrt{a} - \sqrt{\frac{a b^{4}}{4 a}} + \sqrt{a b^{2}}

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

Step 1.6

Reduce the expression

\frac{a b^{4}}{4 a}

$\frac{a b^{4}}{4 a}$ by cancelling the common factors.

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Step 1.6.1

Cancel the common factor.

4 \sqrt{a} - \sqrt{\frac{a b^{4}}{4 a}} + \sqrt{a b^{2}}

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

Step 1.6.2

Rewrite the expression.

4 \sqrt{a} - \sqrt{\frac{b^{4}}{4}} + \sqrt{a b^{2}}

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

4 \sqrt{a} - \sqrt{\frac{b^{4}}{4}} + \sqrt{a b^{2}}

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

Step 1.7

Rewrite

b^{4}

$b^{4}$ as

{(b^{2})}^{2}

${(b^{2})}^{2}$ .

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

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

Step 1.8

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 1.9

Rewrite

\frac{{(b^{2})}^{2}}{2^{2}}

$\frac{{(b^{2})}^{2}}{2^{2}}$ as

{(\frac{b^{2}}{2})}^{2}

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

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

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

Step 1.10

Pull terms out from under the radical, assuming positive real numbers.

4 \sqrt{a} - \frac{b^{2}}{2} + \sqrt{a b^{2}}

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

Step 1.11

Reorder

a

$a$ and

b^{2}

$b^{2}$ .

4 \sqrt{a} - \frac{b^{2}}{2} + \sqrt{b^{2} a}

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

Step 1.12

Pull terms out from under the radical.

4 \sqrt{a} - \frac{b^{2}}{2} + b \sqrt{a}

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

4 \sqrt{a} - \frac{b^{2}}{2} + b \sqrt{a}

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

Step 2

Move

4 \sqrt{a}

$4 \sqrt{a}$ .

- \frac{b^{2}}{2} + b \sqrt{a} + 4 \sqrt{a}

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