Algebra Examples

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

Step 1

Multiply to rationalize the numerator.

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

Step 2

Simplify.

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Step 2.1

Raise $\sqrt{5^{3}}$ to the power of $1$ .

$\frac{2 ({\sqrt{5^{3}}}^{1} \sqrt{5^{3}})}{(6 + 3 \sqrt{10^{5}}) \sqrt{5^{3}}}$

Step 2.2

Raise $\sqrt{5^{3}}$ to the power of $1$ .

$\frac{2 ({\sqrt{5^{3}}}^{1} {\sqrt{5^{3}}}^{1})}{(6 + 3 \sqrt{10^{5}}) \sqrt{5^{3}}}$

Step 2.3

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

$\frac{2 {\sqrt{5^{3}}}^{1 + 1}}{(6 + 3 \sqrt{10^{5}}) \sqrt{5^{3}}}$

Step 2.4

Add $1$ and $1$ .

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

Step 2.5

Rewrite ${\sqrt{5^{3}}}^{2}$ as $5^{3}$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5^{3}}$ as $5^{\frac{3}{2}}$ .

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

Step 2.5.2

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

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

Step 2.5.3

Combine $\frac{3}{2}$ and $2$ .

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

Step 2.5.4

Multiply $3$ by $2$ .

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

Step 2.5.5

Cancel the common factor of $6$ and $2$ .

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Step 2.5.5.1

Factor $2$ out of $6$ .

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

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 5^{\frac{2 \cdot 3}{2 (1)}}}{(6 + 3 \sqrt{10^{5}}) \sqrt{5^{3}}}$

Step 2.5.5.2.2

Cancel the common factor.

$\frac{2 \cdot 5^{\frac{2 \cdot 3}{2 \cdot 1}}}{(6 + 3 \sqrt{10^{5}}) \sqrt{5^{3}}}$

Step 2.5.5.2.3

Rewrite the expression.

$\frac{2 \cdot 5^{\frac{3}{1}}}{(6 + 3 \sqrt{10^{5}}) \sqrt{5^{3}}}$

Step 2.5.5.2.4

Divide $3$ by $1$ .

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

Step 2.6

Simplify each term.

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Step 2.6.1

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

$\frac{2 \cdot 5^{3}}{(6 + 3 \sqrt{100000}) \sqrt{5^{3}}}$

Step 2.6.2

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

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Step 2.6.2.1

Factor $10000$ out of $100000$ .

$\frac{2 \cdot 5^{3}}{(6 + 3 \sqrt{10000 (10)}) \sqrt{5^{3}}}$

Step 2.6.2.2

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

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

Step 2.6.3

Pull terms out from under the radical.

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

Step 2.6.4

Multiply $100$ by $3$ .

$\frac{2 \cdot 5^{3}}{(6 + 300 \sqrt{10}) \sqrt{5^{3}}}$

Step 2.7

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

$\frac{2 \cdot 5^{3}}{(6 + 300 \sqrt{10}) \sqrt{125}}$

Step 2.8

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

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Step 2.8.1

Factor $25$ out of $125$ .

$\frac{2 \cdot 5^{3}}{(6 + 300 \sqrt{10}) \sqrt{25 (5)}}$

Step 2.8.2

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

$\frac{2 \cdot 5^{3}}{(6 + 300 \sqrt{10}) \sqrt{5^{2} \cdot 5}}$

Step 2.9

Pull terms out from under the radical.

$\frac{2 \cdot 5^{3}}{(6 + 300 \sqrt{10}) (5 \sqrt{5})}$

Step 2.10

Apply the distributive property.

$\frac{2 \cdot 5^{3}}{6 (5 \sqrt{5}) + 300 \sqrt{10} (5 \sqrt{5})}$

Step 2.11

Multiply $5$ by $6$ .

$\frac{2 \cdot 5^{3}}{30 \sqrt{5} + 300 \sqrt{10} (5 \sqrt{5})}$

Step 2.12

Multiply $300 \sqrt{10} (5 \sqrt{5})$ .

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Step 2.12.1

Multiply $5$ by $300$ .

$\frac{2 \cdot 5^{3}}{30 \sqrt{5} + 1500 \sqrt{10} \sqrt{5}}$

Step 2.12.2

Combine using the product rule for radicals.

$\frac{2 \cdot 5^{3}}{30 \sqrt{5} + 1500 \sqrt{5 \cdot 10}}$

Step 2.12.3

Multiply $5$ by $10$ .

$\frac{2 \cdot 5^{3}}{30 \sqrt{5} + 1500 \sqrt{50}}$

Step 2.13

Simplify each term.

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Step 2.13.1

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

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Step 2.13.1.1

Factor $25$ out of $50$ .

$\frac{2 \cdot 5^{3}}{30 \sqrt{5} + 1500 \sqrt{25 (2)}}$

Step 2.13.1.2

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

$\frac{2 \cdot 5^{3}}{30 \sqrt{5} + 1500 \sqrt{5^{2} \cdot 2}}$

Step 2.13.2

Pull terms out from under the radical.

$\frac{2 \cdot 5^{3}}{30 \sqrt{5} + 1500 (5 \sqrt{2})}$

Step 2.13.3

Multiply $5$ by $1500$ .

$\frac{2 \cdot 5^{3}}{30 \sqrt{5} + 7500 \sqrt{2}}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \cdot 5^{3}}{30 \sqrt{5} + 7500 \sqrt{2}}$

Decimal Form:

$0.02342209 \dots$