Basic Math Examples

4 \sqrt{80} \cdot \sqrt{20}

$4 \sqrt{80} \cdot \sqrt{20}$

Step 1

Rewrite

80

$80$ as

4^{2} \cdot 5

$4^{2} \cdot 5$ .

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Step 1.1

Factor

16

$16$ out of

80

$80$ .

4 \sqrt{16 (5)} \cdot \sqrt{20}

$4 \sqrt{16 (5)} \cdot \sqrt{20}$

Step 1.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

4 \sqrt{4^{2} \cdot 5} \cdot \sqrt{20}

$4 \sqrt{4^{2} \cdot 5} \cdot \sqrt{20}$

4 \sqrt{4^{2} \cdot 5} \cdot \sqrt{20}

$4 \sqrt{4^{2} \cdot 5} \cdot \sqrt{20}$

Step 2

Pull terms out from under the radical.

4 (4 \sqrt{5}) \cdot \sqrt{20}

$4 (4 \sqrt{5}) \cdot \sqrt{20}$

Step 3

Multiply

4

$4$ by

4

$4$ .

16 \sqrt{5} \cdot \sqrt{20}

$16 \sqrt{5} \cdot \sqrt{20}$

Step 4

Rewrite

20

$20$ as

2^{2} \cdot 5

$2^{2} \cdot 5$ .

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Step 4.1

Factor

4

$4$ out of

20

$20$ .

16 \sqrt{5} \cdot \sqrt{4 (5)}

$16 \sqrt{5} \cdot \sqrt{4 (5)}$

Step 4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

16 \sqrt{5} \cdot \sqrt{2^{2} \cdot 5}

$16 \sqrt{5} \cdot \sqrt{2^{2} \cdot 5}$

16 \sqrt{5} \cdot \sqrt{2^{2} \cdot 5}

$16 \sqrt{5} \cdot \sqrt{2^{2} \cdot 5}$

Step 5

Pull terms out from under the radical.

16 \sqrt{5} \cdot (2 \sqrt{5})

$16 \sqrt{5} \cdot (2 \sqrt{5})$

Step 6

Multiply

16 \sqrt{5} (2 \sqrt{5})

$16 \sqrt{5} (2 \sqrt{5})$ .

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Step 6.1

Multiply

2

$2$ by

16

$16$ .

32 \sqrt{5} \sqrt{5}

$32 \sqrt{5} \sqrt{5}$

Step 6.2

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

32 ({\sqrt{5}}^{1} \sqrt{5})

$32 ({\sqrt{5}}^{1} \sqrt{5})$

Step 6.3

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

32 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})

$32 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

Step 6.4

Use the power rule

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

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

32 {\sqrt{5}}^{1 + 1}

$32 {\sqrt{5}}^{1 + 1}$

Step 6.5

Add

1

$1$ and

1

$1$ .

32 {\sqrt{5}}^{2}

$32 {\sqrt{5}}^{2}$

32 {\sqrt{5}}^{2}

$32 {\sqrt{5}}^{2}$

Step 7

Rewrite

{\sqrt{5}}^{2}

${\sqrt{5}}^{2}$ as

5

$5$ .

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Step 7.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{5}

$\sqrt{5}$ as

5^{\frac{1}{2}}

$5^{\frac{1}{2}}$ .

32 {(5^{\frac{1}{2}})}^{2}

$32 {(5^{\frac{1}{2}})}^{2}$

Step 7.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

32 \cdot 5^{\frac{1}{2} \cdot 2}

$32 \cdot 5^{\frac{1}{2} \cdot 2}$

Step 7.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

32 \cdot 5^{\frac{2}{2}}

$32 \cdot 5^{\frac{2}{2}}$

Step 7.4

Cancel the common factor of

2

$2$ .

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Step 7.4.1

Cancel the common factor.

$32 \cdot 5^{\frac{2}{2}}$

Step 7.4.2

Rewrite the expression.

$32 \cdot 5^{1}$

$32 \cdot 5^{1}$

Step 7.5

Evaluate the exponent.

$32 \cdot 5$

$32 \cdot 5$

Step 8

Multiply

$32$ by

$5$ .

$160$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$