Algebra Examples

$\frac{\sqrt{15 x^{6}}}{\sqrt{5 x^{5}}} \cdot \sqrt{3 x^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $15 x^{6}$ as ${(x^{3})}^{2} \cdot 15$ .

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Step 1.1.1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$\frac{\sqrt{15 {(x^{3})}^{2}}}{\sqrt{5 x^{5}}} \cdot \sqrt{3 x^{3}}$

Step 1.1.2

Reorder $15$ and ${(x^{3})}^{2}$ .

$\frac{\sqrt{{(x^{3})}^{2} \cdot 15}}{\sqrt{5 x^{5}}} \cdot \sqrt{3 x^{3}}$

Step 1.2

Pull terms out from under the radical.

$\frac{x^{3} \sqrt{15}}{\sqrt{5 x^{5}}} \cdot \sqrt{3 x^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $5 x^{5}$ as ${(x^{2})}^{2} \cdot (5 x)$ .

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Step 2.1.1

Factor out $x^{4}$ .

$\frac{x^{3} \sqrt{15}}{\sqrt{5 (x^{4} x)}} \cdot \sqrt{3 x^{3}}$

Step 2.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{x^{3} \sqrt{15}}{\sqrt{5 ({(x^{2})}^{2} x)}} \cdot \sqrt{3 x^{3}}$

Step 2.1.3

Reorder $5$ and ${(x^{2})}^{2}$ .

$\frac{x^{3} \sqrt{15}}{\sqrt{{(x^{2})}^{2} \cdot 5 x}} \cdot \sqrt{3 x^{3}}$

Step 2.1.4

Add parentheses.

$\frac{x^{3} \sqrt{15}}{\sqrt{{(x^{2})}^{2} \cdot (5 x)}} \cdot \sqrt{3 x^{3}}$

Step 2.2

Pull terms out from under the radical.

$\frac{x^{3} \sqrt{15}}{x^{2} \sqrt{5 x}} \cdot \sqrt{3 x^{3}}$

Step 3

Cancel the common factor of $x^{3}$ and $x^{2}$ .

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Step 3.1

Factor $x^{2}$ out of $x^{3} \sqrt{15}$ .

$\frac{x^{2} (x \sqrt{15})}{x^{2} \sqrt{5 x}} \cdot \sqrt{3 x^{3}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\frac{x^{2} (x \sqrt{15})}{x^{2} (\sqrt{5 x})} \cdot \sqrt{3 x^{3}}$

Step 3.2.2

Cancel the common factor.

$\frac{x^{2} (x \sqrt{15})}{x^{2} \sqrt{5 x}} \cdot \sqrt{3 x^{3}}$

Step 3.2.3

Rewrite the expression.

$\frac{x \sqrt{15}}{\sqrt{5 x}} \cdot \sqrt{3 x^{3}}$

Step 4

Combine $\sqrt{15}$ and $\sqrt{5 x}$ into a single radical.

$x \sqrt{\frac{15}{5 x}} \cdot \sqrt{3 x^{3}}$

Step 5

Reduce the expression $\frac{15}{5 x}$ by cancelling the common factors.

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Step 5.1

Factor $5$ out of $15$ .

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

Step 5.2

Factor $5$ out of $5 x$ .

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

Step 5.3

Cancel the common factor.

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

Step 5.4

Rewrite the expression.

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

Step 6

Rewrite $\sqrt{\frac{3}{x}}$ as $\frac{\sqrt{3}}{\sqrt{x}}$ .

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

$x \frac{\sqrt{3} \sqrt{x}}{{\sqrt{x}}^{1} \sqrt{x}} \cdot \sqrt{3 x^{3}}$

Step 8.3

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

$x \frac{\sqrt{3} \sqrt{x}}{{\sqrt{x}}^{1} {\sqrt{x}}^{1}} \cdot \sqrt{3 x^{3}}$

Step 8.4

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

$x \frac{\sqrt{3} \sqrt{x}}{{\sqrt{x}}^{1 + 1}} \cdot \sqrt{3 x^{3}}$

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

$x \frac{\sqrt{3} \sqrt{x}}{x^{1}} \cdot \sqrt{3 x^{3}}$

Step 8.6.5

Simplify.

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

Step 9

Combine using the product rule for radicals.

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

Step 10

Cancel the common factor of $x$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

$\sqrt{3 x} \cdot \sqrt{3 x^{3}}$

Step 11

Rewrite $3 x^{3}$ as $x^{2} \cdot (3 x)$ .

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Step 11.1

Factor out $x^{2}$ .

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

Step 11.2

Reorder $3$ and $x^{2}$ .

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

Step 11.3

Add parentheses.

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

Step 12

Pull terms out from under the radical.

$\sqrt{3 x} \cdot (x \sqrt{3 x})$

Step 13

Multiply $\sqrt{3 x} (x \sqrt{3 x})$ .

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Step 13.1

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

$x ({\sqrt{3 x}}^{1} \sqrt{3 x})$

Step 13.2

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

$x ({\sqrt{3 x}}^{1} {\sqrt{3 x}}^{1})$

Step 13.3

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

$x {\sqrt{3 x}}^{1 + 1}$

Step 13.4

Add $1$ and $1$ .

$x {\sqrt{3 x}}^{2}$

Step 14

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

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Step 14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x}$ as ${(3 x)}^{\frac{1}{2}}$ .

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

Step 14.2

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

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

Step 14.3

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

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

Step 14.4

Cancel the common factor of $2$ .

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Step 14.4.1

Cancel the common factor.

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

Step 14.4.2

Rewrite the expression.

$x {(3 x)}^{1}$

Step 14.5

Simplify.

$x (3 x)$

Step 15

Rewrite using the commutative property of multiplication.

$3 x \cdot x$

Step 16

Multiply $x$ by $x$ by adding the exponents.

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Step 16.1

Move $x$ .

$3 (x \cdot x)$

Step 16.2

Multiply $x$ by $x$ .

$3 x^{2}$