Algebra Examples

$\frac{6 \sqrt{18}}{12 \sqrt{40}}$

Step 1

Cancel the common factor of $6$ and $12$ .

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Step 1.1

Factor $6$ out of $6 \sqrt{18}$ .

$\frac{6 (\sqrt{18})}{12 \sqrt{40}}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $6$ out of $12 \sqrt{40}$ .

$\frac{6 (\sqrt{18})}{6 (2 \sqrt{40})}$

Step 1.2.2

Cancel the common factor.

$\frac{6 \sqrt{18}}{6 (2 \sqrt{40})}$

Step 1.2.3

Rewrite the expression.

$\frac{\sqrt{18}}{2 \sqrt{40}}$

Step 2

Combine $\sqrt{18}$ and $\sqrt{40}$ into a single radical.

$\frac{\sqrt{\frac{18}{40}}}{2}$

Step 3

Cancel the common factor of $18$ and $40$ .

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Step 3.1

Factor $2$ out of $18$ .

$\frac{\sqrt{\frac{2 (9)}{40}}}{2}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $40$ .

$\frac{\sqrt{\frac{2 \cdot 9}{2 \cdot 20}}}{2}$

Step 3.2.2

Cancel the common factor.

$\frac{\sqrt{\frac{2 \cdot 9}{2 \cdot 20}}}{2}$

Step 3.2.3

Rewrite the expression.

$\frac{\sqrt{\frac{9}{20}}}{2}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $\sqrt{\frac{9}{20}}$ as $\frac{\sqrt{9}}{\sqrt{20}}$ .

$\frac{\frac{\sqrt{9}}{\sqrt{20}}}{2}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

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

$\frac{\frac{\sqrt{3^{2}}}{\sqrt{20}}}{2}$

Step 4.2.2

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

$\frac{\frac{3}{\sqrt{20}}}{2}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

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

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Step 4.3.1.1

Factor $4$ out of $20$ .

$\frac{\frac{3}{\sqrt{4 (5)}}}{2}$

Step 4.3.1.2

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

$\frac{\frac{3}{\sqrt{2^{2} \cdot 5}}}{2}$

Step 4.3.2

Pull terms out from under the radical.

$\frac{\frac{3}{2 \sqrt{5}}}{2}$

Step 4.4

Multiply $\frac{3}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\frac{3}{2 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}}{2}$

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

Multiply $\frac{3}{2 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\frac{3 \sqrt{5}}{2 \sqrt{5} \sqrt{5}}}{2}$

Step 4.5.2

Move $\sqrt{5}$ .

$\frac{\frac{3 \sqrt{5}}{2 (\sqrt{5} \sqrt{5})}}{2}$

Step 4.5.3

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

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

Step 4.5.4

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

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

Step 4.5.5

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

$\frac{\frac{3 \sqrt{5}}{2 {\sqrt{5}}^{1 + 1}}}{2}$

Step 4.5.6

Add $1$ and $1$ .

$\frac{\frac{3 \sqrt{5}}{2 {\sqrt{5}}^{2}}}{2}$

Step 4.5.7

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

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Step 4.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 4.5.7.2

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

$\frac{\frac{3 \sqrt{5}}{2 \cdot 5^{\frac{1}{2} \cdot 2}}}{2}$

Step 4.5.7.3

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

$\frac{\frac{3 \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}}{2}$

Step 4.5.7.4

Cancel the common factor of $2$ .

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Step 4.5.7.4.1

Cancel the common factor.

$\frac{\frac{3 \sqrt{5}}{2 \cdot 5^{\frac{2}{2}}}}{2}$

Step 4.5.7.4.2

Rewrite the expression.

$\frac{\frac{3 \sqrt{5}}{2 \cdot 5^{1}}}{2}$

Step 4.5.7.5

Evaluate the exponent.

$\frac{\frac{3 \sqrt{5}}{2 \cdot 5}}{2}$

Step 4.6

Multiply $2$ by $5$ .

$\frac{\frac{3 \sqrt{5}}{10}}{2}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 \sqrt{5}}{10} \cdot \frac{1}{2}$

Step 6

Multiply $\frac{3 \sqrt{5}}{10} \cdot \frac{1}{2}$ .

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Step 6.1

Multiply $\frac{3 \sqrt{5}}{10}$ by $\frac{1}{2}$ .

$\frac{3 \sqrt{5}}{10 \cdot 2}$

Step 6.2

Multiply $10$ by $2$ .

$\frac{3 \sqrt{5}}{20}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{5}}{20}$

Decimal Form:

$0.33541019 \dots$