Algebra Examples

$\frac{2 \sqrt{5^{3}}}{6 + 3 \sqrt{10^{5}}}$

Step 1

Simplify the numerator.

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Step 1.1

Raise $5$ to the power of $3$ .

$\frac{2 \sqrt{125}}{6 + 3 \sqrt{10^{5}}}$

Step 1.2

Rewrite $125$ as $5^{2} \cdot 5$ .

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Step 1.2.1

Factor $25$ out of $125$ .

$\frac{2 \sqrt{25 (5)}}{6 + 3 \sqrt{10^{5}}}$

Step 1.2.2

Rewrite $25$ as $5^{2}$ .

$\frac{2 \sqrt{5^{2} \cdot 5}}{6 + 3 \sqrt{10^{5}}}$

Step 1.3

Pull terms out from under the radical.

$\frac{2 \cdot 5 \sqrt{5}}{6 + 3 \sqrt{10^{5}}}$

Step 1.4

Multiply $2$ by $5$ .

$\frac{10 \sqrt{5}}{6 + 3 \sqrt{10^{5}}}$

Step 2

Simplify the denominator.

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Step 2.1

Raise $10$ to the power of $5$ .

$\frac{10 \sqrt{5}}{6 + 3 \sqrt{100000}}$

Step 2.2

Rewrite $100000$ as $100^{2} \cdot 10$ .

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Step 2.2.1

Factor $10000$ out of $100000$ .

$\frac{10 \sqrt{5}}{6 + 3 \sqrt{10000 (10)}}$

Step 2.2.2

Rewrite $10000$ as $100^{2}$ .

$\frac{10 \sqrt{5}}{6 + 3 \sqrt{100^{2} \cdot 10}}$

Step 2.3

Pull terms out from under the radical.

$\frac{10 \sqrt{5}}{6 + 3 (100 \sqrt{10})}$

Step 2.4

Multiply $100$ by $3$ .

$\frac{10 \sqrt{5}}{6 + 300 \sqrt{10}}$

Step 3

Cancel the common factor of $10$ and $6 + 300 \sqrt{10}$ .

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Step 3.1

Factor $2$ out of $10 \sqrt{5}$ .

$\frac{2 (5 \sqrt{5})}{6 + 300 \sqrt{10}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $6$ .

$\frac{2 (5 \sqrt{5})}{2 (3) + 300 \sqrt{10}}$

Step 3.2.2

Factor $2$ out of $300 \sqrt{10}$ .

$\frac{2 (5 \sqrt{5})}{2 (3) + 2 (150 \sqrt{10})}$

Step 3.2.3

Factor $2$ out of $2 (3) + 2 (150 \sqrt{10})$ .

$\frac{2 (5 \sqrt{5})}{2 (3 + 150 \sqrt{10})}$

Step 3.2.4

Cancel the common factor.

$\frac{2 (5 \sqrt{5})}{2 (3 + 150 \sqrt{10})}$

Step 3.2.5

Rewrite the expression.

$\frac{5 \sqrt{5}}{3 + 150 \sqrt{10}}$

Step 4

Multiply $\frac{5 \sqrt{5}}{3 + 150 \sqrt{10}}$ by $\frac{3 - 150 \sqrt{10}}{3 - 150 \sqrt{10}}$ .

$\frac{5 \sqrt{5}}{3 + 150 \sqrt{10}} \cdot \frac{3 - 150 \sqrt{10}}{3 - 150 \sqrt{10}}$

Step 5

Multiply $\frac{5 \sqrt{5}}{3 + 150 \sqrt{10}}$ by $\frac{3 - 150 \sqrt{10}}{3 - 150 \sqrt{10}}$ .

$\frac{5 \sqrt{5} (3 - 150 \sqrt{10})}{(3 + 150 \sqrt{10}) (3 - 150 \sqrt{10})}$

Step 6

Expand the denominator using the FOIL method.

$\frac{5 \sqrt{5} (3 - 150 \sqrt{10})}{9 - 450 \sqrt{10} + 450 \sqrt{10} - 22500 {\sqrt{10}}^{2}}$

Step 7

Simplify.

$\frac{5 \sqrt{5} (3 - 150 \sqrt{10})}{- 224991}$

Step 8

Cancel the common factor of $3 - 150 \sqrt{10}$ and $- 224991$ .

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Step 8.1

Factor $3$ out of $5 \sqrt{5} (3 - 150 \sqrt{10})$ .

$\frac{3 (5 \sqrt{5} (1 - 50 \sqrt{10}))}{- 224991}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $3$ out of $- 224991$ .

$\frac{3 (5 \sqrt{5} (1 - 50 \sqrt{10}))}{3 (- 74997)}$

Step 8.2.2

Cancel the common factor.

$\frac{3 (5 \sqrt{5} (1 - 50 \sqrt{10}))}{3 \cdot - 74997}$

Step 8.2.3

Rewrite the expression.

$\frac{5 \sqrt{5} (1 - 50 \sqrt{10})}{- 74997}$

Step 9

Group $1 - 50 \sqrt{10}$ and $\sqrt{5}$ together.

$\frac{5 ((1 - 50 \sqrt{10}) \sqrt{5})}{- 74997}$

Step 10

Apply the distributive property.

$\frac{5 (1 \sqrt{5} - 50 \sqrt{10} \sqrt{5})}{- 74997}$

Step 11

Multiply $\sqrt{5}$ by $1$ .

$\frac{5 (\sqrt{5} - 50 \sqrt{10} \sqrt{5})}{- 74997}$

Step 12

Multiply $- 50 \sqrt{10} \sqrt{5}$ .

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Step 12.1

Combine using the product rule for radicals.

$\frac{5 (\sqrt{5} - 50 \sqrt{5 \cdot 10})}{- 74997}$

Step 12.2

Multiply $5$ by $10$ .

$\frac{5 (\sqrt{5} - 50 \sqrt{50})}{- 74997}$

Step 13

Simplify each term.

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Step 13.1

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 13.1.1

Factor $25$ out of $50$ .

$\frac{5 (\sqrt{5} - 50 \sqrt{25 (2)})}{- 74997}$

Step 13.1.2

Rewrite $25$ as $5^{2}$ .

$\frac{5 (\sqrt{5} - 50 \sqrt{5^{2} \cdot 2})}{- 74997}$

Step 13.2

Pull terms out from under the radical.

$\frac{5 (\sqrt{5} - 50 (5 \sqrt{2}))}{- 74997}$

Step 13.3

Multiply $5$ by $- 50$ .

$\frac{5 (\sqrt{5} - 250 \sqrt{2})}{- 74997}$

Step 14

Move the negative in front of the fraction.

$- \frac{5 (\sqrt{5} - 250 \sqrt{2})}{74997}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$- \frac{5 (\sqrt{5} - 250 \sqrt{2})}{74997}$

Decimal Form:

$0.02342209 \dots$