Pre-Algebra Examples

$\frac{\sqrt{125 + \sqrt{80}}}{\sqrt{180 - \sqrt{45}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Factor $16$ out of $80$ .

$\frac{\sqrt{125 + \sqrt{16 (5)}}}{\sqrt{180 - \sqrt{45}}}$

Step 1.1.2

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

$\frac{\sqrt{125 + \sqrt{4^{2} \cdot 5}}}{\sqrt{180 - \sqrt{45}}}$

Step 1.2

Pull terms out from under the radical.

$\frac{\sqrt{125 + 4 \sqrt{5}}}{\sqrt{180 - \sqrt{45}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 2.1.1

Factor $9$ out of $45$ .

$\frac{\sqrt{125 + 4 \sqrt{5}}}{\sqrt{180 - \sqrt{9 (5)}}}$

Step 2.1.2

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

$\frac{\sqrt{125 + 4 \sqrt{5}}}{\sqrt{180 - \sqrt{3^{2} \cdot 5}}}$

Step 2.2

Pull terms out from under the radical.

$\frac{\sqrt{125 + 4 \sqrt{5}}}{\sqrt{180 - (3 \sqrt{5})}}$

Step 2.3

Multiply $3$ by $- 1$ .

$\frac{\sqrt{125 + 4 \sqrt{5}}}{\sqrt{180 - 3 \sqrt{5}}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{125 + 4 \sqrt{5}}}{\sqrt{180 - 3 \sqrt{5}}}$

Decimal Form:

$0.87917042 \dots$