Algebra Examples

$\frac{\sqrt{125}}{\sqrt{18} \cdot \sqrt{50}}$

Step 1

Combine $\sqrt{125}$ and $\sqrt{50}$ into a single radical.

$\frac{\sqrt{\frac{125}{50}}}{\sqrt{18}}$

Step 2

Cancel the common factor of $125$ and $50$ .

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Step 2.1

Factor $25$ out of $125$ .

$\frac{\sqrt{\frac{25 (5)}{50}}}{\sqrt{18}}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $25$ out of $50$ .

$\frac{\sqrt{\frac{25 \cdot 5}{25 \cdot 2}}}{\sqrt{18}}$

Step 2.2.2

Cancel the common factor.

$\frac{\sqrt{\frac{25 \cdot 5}{25 \cdot 2}}}{\sqrt{18}}$

Step 2.2.3

Rewrite the expression.

$\frac{\sqrt{\frac{5}{2}}}{\sqrt{18}}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{\frac{\sqrt{5}}{\sqrt{2}}}{\sqrt{18}}$

Step 3.2

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{5}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}{\sqrt{18}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{5} \sqrt{2}}{\sqrt{2} \sqrt{2}}}{\sqrt{18}}$

Step 3.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}{\sqrt{18}}$

Step 3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}{\sqrt{18}}$

Step 3.3.4

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

$\frac{\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}}{\sqrt{18}}$

Step 3.3.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{2}}}{\sqrt{18}}$

Step 3.3.6

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

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Step 3.3.6.1

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

$\frac{\frac{\sqrt{5} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}{\sqrt{18}}$

Step 3.3.6.2

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

$\frac{\frac{\sqrt{5} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}{\sqrt{18}}$

Step 3.3.6.3

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

$\frac{\frac{\sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}}}{\sqrt{18}}$

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{5} \sqrt{2}}{2^{\frac{2}{2}}}}{\sqrt{18}}$

Step 3.3.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{5} \sqrt{2}}{2^{1}}}{\sqrt{18}}$

Step 3.3.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{5} \sqrt{2}}{2}}{\sqrt{18}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{5 \cdot 2}}{2}}{\sqrt{18}}$

Step 3.4.2

Multiply $5$ by $2$ .

$\frac{\frac{\sqrt{10}}{2}}{\sqrt{18}}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.1.1

Factor $9$ out of $18$ .

$\frac{\frac{\sqrt{10}}{2}}{\sqrt{9 (2)}}$

Step 4.1.2

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

$\frac{\frac{\sqrt{10}}{2}}{\sqrt{3^{2} \cdot 2}}$

Step 4.2

Pull terms out from under the radical.

$\frac{\frac{\sqrt{10}}{2}}{3 \sqrt{2}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{10}}{2} \cdot \frac{1}{3 \sqrt{2}}$

Step 6

Multiply $\frac{1}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{10}}{2} (\frac{1}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{1}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{3 \sqrt{2} \sqrt{2}}$

Step 7.2

Move $\sqrt{2}$ .

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{3 (\sqrt{2} \sqrt{2})}$

Step 7.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.4

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 7.5

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

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{3 {\sqrt{2}}^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{3 {\sqrt{2}}^{2}}$

Step 7.7

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

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Step 7.7.1

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

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

Step 7.7.2

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

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 7.7.3

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

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 7.7.4

Cancel the common factor of $2$ .

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Step 7.7.4.1

Cancel the common factor.

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 7.7.4.2

Rewrite the expression.

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{3 \cdot 2^{1}}$

Step 7.7.5

Evaluate the exponent.

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{3 \cdot 2}$

Step 8

Multiply $3$ by $2$ .

$\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{6}$

Step 9

Multiply $\frac{\sqrt{10}}{2} \cdot \frac{\sqrt{2}}{6}$ .

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Step 9.1

Multiply $\frac{\sqrt{10}}{2}$ by $\frac{\sqrt{2}}{6}$ .

$\frac{\sqrt{10} \sqrt{2}}{2 \cdot 6}$

Step 9.2

Combine using the product rule for radicals.

$\frac{\sqrt{10 \cdot 2}}{2 \cdot 6}$

Step 9.3

Multiply $10$ by $2$ .

$\frac{\sqrt{20}}{2 \cdot 6}$

Step 9.4

Multiply $2$ by $6$ .

$\frac{\sqrt{20}}{12}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 10.1.1

Factor $4$ out of $20$ .

$\frac{\sqrt{4 (5)}}{12}$

Step 10.1.2

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

$\frac{\sqrt{2^{2} \cdot 5}}{12}$

Step 10.2

Pull terms out from under the radical.

$\frac{2 \sqrt{5}}{12}$

Step 11

Cancel the common factor of $2$ and $12$ .

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Step 11.1

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

$\frac{2 (\sqrt{5})}{12}$

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $2$ out of $12$ .

$\frac{2 \sqrt{5}}{2 \cdot 6}$

Step 11.2.2

Cancel the common factor.

$\frac{2 \sqrt{5}}{2 \cdot 6}$

Step 11.2.3

Rewrite the expression.

$\frac{\sqrt{5}}{6}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{5}}{6}$

Decimal Form:

$0.37267799 \dots$