Basic Math Examples

$5 \cdot \sqrt{\frac{80}{25}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 1

Cancel the common factor of $80$ and $25$ .

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Step 1.1

Factor $5$ out of $80$ .

$5 \cdot \sqrt{\frac{5 (16)}{25}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $5$ out of $25$ .

$5 \cdot \sqrt{\frac{5 \cdot 16}{5 \cdot 5}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 1.2.2

Cancel the common factor.

$5 \cdot \sqrt{\frac{5 \cdot 16}{5 \cdot 5}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 1.2.3

Rewrite the expression.

$5 \cdot \sqrt{\frac{16}{5}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 2

Rewrite $\sqrt{\frac{16}{5}}$ as $\frac{\sqrt{16}}{\sqrt{5}}$ .

$5 \cdot \frac{\sqrt{16}}{\sqrt{5}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$5 \cdot \frac{\sqrt{4^{2}}}{\sqrt{5}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 3.2

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

$5 \cdot \frac{4}{\sqrt{5}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 4

Multiply $\frac{4}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$5 \cdot (\frac{4}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}) + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{4}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$5 \cdot \frac{4 \sqrt{5}}{\sqrt{5} \sqrt{5}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.2

Raise $\sqrt{5}$ to the power of $1$ .

$5 \cdot \frac{4 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.3

Raise $\sqrt{5}$ to the power of $1$ .

$5 \cdot \frac{4 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.4

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

$5 \cdot \frac{4 \sqrt{5}}{{\sqrt{5}}^{1 + 1}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.5

Add $1$ and $1$ .

$5 \cdot \frac{4 \sqrt{5}}{{\sqrt{5}}^{2}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.6

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

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Step 5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$5 \cdot \frac{4 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.6.2

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

$5 \cdot \frac{4 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.6.3

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

$5 \cdot \frac{4 \sqrt{5}}{5^{\frac{2}{2}}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$5 \cdot \frac{4 \sqrt{5}}{5^{\frac{2}{2}}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.6.4.2

Rewrite the expression.

$5 \cdot \frac{4 \sqrt{5}}{5^{1}} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 5.6.5

Evaluate the exponent.

$5 \cdot \frac{4 \sqrt{5}}{5} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 6

Cancel the common factor of $5$ .

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Step 6.1

Cancel the common factor.

$5 \cdot \frac{4 \sqrt{5}}{5} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 6.2

Rewrite the expression.

$4 \sqrt{5} + \frac{80 \sqrt{5}}{\sqrt{80}}$

Step 7

Combine $\sqrt{5}$ and $\sqrt{80}$ into a single radical.

$4 \sqrt{5} + 80 \sqrt{\frac{5}{80}}$

Step 8

Cancel the common factor of $5$ and $80$ .

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Step 8.1

Factor $5$ out of $5$ .

$4 \sqrt{5} + 80 \sqrt{\frac{5 (1)}{80}}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $5$ out of $80$ .

$4 \sqrt{5} + 80 \sqrt{\frac{5 \cdot 1}{5 \cdot 16}}$

Step 8.2.2

Cancel the common factor.

$4 \sqrt{5} + 80 \sqrt{\frac{5 \cdot 1}{5 \cdot 16}}$

Step 8.2.3

Rewrite the expression.

$4 \sqrt{5} + 80 \sqrt{\frac{1}{16}}$

Step 9

Rewrite $\sqrt{\frac{1}{16}}$ as $\frac{\sqrt{1}}{\sqrt{16}}$ .

$4 \sqrt{5} + 80 \frac{\sqrt{1}}{\sqrt{16}}$

Step 10

Any root of $1$ is $1$ .

$4 \sqrt{5} + 80 \frac{1}{\sqrt{16}}$

Step 11

Simplify the denominator.

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Step 11.1

Rewrite $16$ as $4^{2}$ .

$4 \sqrt{5} + 80 \frac{1}{\sqrt{4^{2}}}$

Step 11.2

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

$4 \sqrt{5} + 80 (\frac{1}{4})$

Step 12

Cancel the common factor of $4$ .

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Step 12.1

Factor $4$ out of $80$ .

$4 \sqrt{5} + 4 (20) \frac{1}{4}$

Step 12.2

Cancel the common factor.

$4 \sqrt{5} + 4 \cdot 20 \frac{1}{4}$

Step 12.3

Rewrite the expression.

$4 \sqrt{5} + 20$

Step 13

The result can be shown in multiple forms.

Exact Form:

$4 \sqrt{5} + 20$

Decimal Form:

$28.94427190 \dots$