Pre-Algebra Examples

$\sqrt{\frac{5}{81} + \sqrt{\frac{80}{9}}}$

Step 1

Rewrite $\sqrt{\frac{80}{9}}$ as $\frac{\sqrt{80}}{\sqrt{9}}$ .

$\sqrt{\frac{5}{81} + \frac{\sqrt{80}}{\sqrt{9}}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 2.1.1

Factor $16$ out of $80$ .

$\sqrt{\frac{5}{81} + \frac{\sqrt{16 (5)}}{\sqrt{9}}}$

Step 2.1.2

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

$\sqrt{\frac{5}{81} + \frac{\sqrt{4^{2} \cdot 5}}{\sqrt{9}}}$

Step 2.2

Pull terms out from under the radical.

$\sqrt{\frac{5}{81} + \frac{4 \sqrt{5}}{\sqrt{9}}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $9$ as $3^{2}$ .

$\sqrt{\frac{5}{81} + \frac{4 \sqrt{5}}{\sqrt{3^{2}}}}$

Step 3.2

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

$\sqrt{\frac{5}{81} + \frac{4 \sqrt{5}}{3}}$

Step 4

To write $\frac{4 \sqrt{5}}{3}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\sqrt{\frac{5}{81} + \frac{4 \sqrt{5}}{3} \cdot \frac{27}{27}}$

Step 5

Write each expression with a common denominator of $81$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{4 \sqrt{5}}{3}$ by $\frac{27}{27}$ .

$\sqrt{\frac{5}{81} + \frac{4 \sqrt{5} \cdot 27}{3 \cdot 27}}$

Step 5.2

Multiply $3$ by $27$ .

$\sqrt{\frac{5}{81} + \frac{4 \sqrt{5} \cdot 27}{81}}$

Step 6

Combine the numerators over the common denominator.

$\sqrt{\frac{5 + 4 \sqrt{5} \cdot 27}{81}}$

Step 7

Multiply $27$ by $4$ .

$\sqrt{\frac{5 + 108 \sqrt{5}}{81}}$

Step 8

Rewrite $\sqrt{\frac{5 + 108 \sqrt{5}}{81}}$ as $\frac{\sqrt{5 + 108 \sqrt{5}}}{\sqrt{81}}$ .

$\frac{\sqrt{5 + 108 \sqrt{5}}}{\sqrt{81}}$

Step 9

Simplify the denominator.

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Step 9.1

Rewrite $81$ as $9^{2}$ .

$\frac{\sqrt{5 + 108 \sqrt{5}}}{\sqrt{9^{2}}}$

Step 9.2

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

$\frac{\sqrt{5 + 108 \sqrt{5}}}{9}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5 + 108 \sqrt{5}}}{9}$

Decimal Form:

$1.74446334 \dots$