Algebra Examples

$\frac{x \sqrt{18 x}}{3 \sqrt{5 x^{3}}}$

Step 1

Combine $\sqrt{18 x}$ and $\sqrt{5 x^{3}}$ into a single radical.

$\frac{x \sqrt{\frac{18 x}{5 x^{3}}}}{3}$

Step 2

Cancel the common factor of $x$ and $x^{3}$ .

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Step 2.1

Factor $x$ out of $18 x$ .

$\frac{x \sqrt{\frac{x \cdot 18}{5 x^{3}}}}{3}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $x$ out of $5 x^{3}$ .

$\frac{x \sqrt{\frac{x \cdot 18}{x (5 x^{2})}}}{3}$

Step 2.2.2

Cancel the common factor.

$\frac{x \sqrt{\frac{x \cdot 18}{x (5 x^{2})}}}{3}$

Step 2.2.3

Rewrite the expression.

$\frac{x \sqrt{\frac{18}{5 x^{2}}}}{3}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $\frac{18}{5 x^{2}}$ as ${(\frac{3}{x})}^{2} \frac{2}{5}$ .

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Step 3.1.1

Factor the perfect power $3^{2}$ out of $18$ .

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

Step 3.1.2

Factor the perfect power $x^{2}$ out of $5 x^{2}$ .

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

Step 3.1.3

Rearrange the fraction $\frac{3^{2} \cdot 2}{x^{2} \cdot 5}$ .

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

Step 3.2

Pull terms out from under the radical.

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine and simplify the denominator.

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

Step 3.5.4

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

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

Step 3.5.5

Add $1$ and $1$ .

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

Step 3.5.6

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

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Step 3.5.6.1

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

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

Step 3.5.6.2

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

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

Step 3.5.6.3

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

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

Step 3.5.6.4

Cancel the common factor of $2$ .

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Step 3.5.6.4.1

Cancel the common factor.

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

Step 3.5.6.4.2

Rewrite the expression.

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

Step 3.5.6.5

Evaluate the exponent.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Combine using the product rule for radicals.

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

Step 3.6.2

Multiply $2$ by $5$ .

$\frac{x (\frac{3}{x} \cdot \frac{\sqrt{10}}{5})}{3}$

Step 3.7

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

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

Step 3.8

Move $5$ to the left of $x$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine $x$ and $\frac{3 \sqrt{10}}{5 x}$ .

$\frac{\frac{x (3 \sqrt{10})}{5 x}}{3}$

Step 4.2

Reduce the expression $\frac{x \cdot 3 \sqrt{10}}{5 x}$ by cancelling the common factors.

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $3$ .

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Step 6.1

Factor $3$ out of $3 \sqrt{10}$ .

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.63245553 \dots$