Algebra Examples

$\frac{\sqrt{5 x^{7}}}{\sqrt{3 x}} \cdot \sqrt{15 x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $5 x^{7}$ as ${(x^{3})}^{2} \cdot (5 x)$ .

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Step 1.1.1

Factor out $x^{6}$ .

$\frac{\sqrt{5 (x^{6} x)}}{\sqrt{3 x}} \cdot \sqrt{15 x^{2}}$

Step 1.1.2

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

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

Step 1.1.3

Reorder $5$ and ${(x^{3})}^{2}$ .

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

Step 1.1.4

Add parentheses.

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Rewrite $\sqrt{\frac{5}{3}}$ as $\frac{\sqrt{5}}{\sqrt{3}}$ .

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

$x^{3} \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot \sqrt{15 x^{2}}$

Step 6.3

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

$x^{3} \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot \sqrt{15 x^{2}}$

Step 6.4

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

$x^{3} \frac{\sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot \sqrt{15 x^{2}}$

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

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

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Step 6.6.1

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

$x^{3} \frac{\sqrt{5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot \sqrt{15 x^{2}}$

Step 6.6.2

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

$x^{3} \frac{\sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot \sqrt{15 x^{2}}$

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

$x^{3} \frac{\sqrt{5} \sqrt{3}}{3^{1}} \cdot \sqrt{15 x^{2}}$

Step 6.6.5

Evaluate the exponent.

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

Step 7

Simplify the numerator.

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $5$ by $3$ .

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

Step 8

Simplify terms.

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Step 8.1

Combine $x^{3}$ and $\frac{\sqrt{15}}{3}$ .

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

Step 8.2

Reorder $15$ and $x^{2}$ .

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

Step 8.3

Pull terms out from under the radical.

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

Step 8.4

Rewrite using the commutative property of multiplication.

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

Step 9

Multiply $\sqrt{15} \frac{x^{3} \sqrt{15}}{3}$ .

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Step 9.1

Combine $\sqrt{15}$ and $\frac{x^{3} \sqrt{15}}{3}$ .

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

Step 9.2

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{x^{3} ({\sqrt{15}}^{1} \sqrt{15})}{3} x$

Step 9.3

Raise $\sqrt{15}$ to the power of $1$ .

$\frac{x^{3} ({\sqrt{15}}^{1} {\sqrt{15}}^{1})}{3} x$

Step 9.4

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

$\frac{x^{3} {\sqrt{15}}^{1 + 1}}{3} x$

Step 9.5

Add $1$ and $1$ .

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

Step 10

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\frac{x^{3} {(15^{\frac{1}{2}})}^{2}}{3} x$

Step 10.2

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

$\frac{x^{3} \cdot 15^{\frac{1}{2} \cdot 2}}{3} x$

Step 10.3

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

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

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Cancel the common factor.

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

Step 10.4.2

Rewrite the expression.

$\frac{x^{3} \cdot 15^{1}}{3} x$

Step 10.5

Evaluate the exponent.

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

Step 11

Cancel the common factor of $15$ and $3$ .

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Step 11.1

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

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

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $3$ out of $3$ .

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

Step 11.2.2

Cancel the common factor.

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

Step 11.2.3

Rewrite the expression.

$\frac{x^{3} \cdot 5}{1} x$

Step 11.2.4

Divide $x^{3} \cdot 5$ by $1$ .

$x^{3} \cdot 5 x$

Step 12

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 12.1

Move $x$ .

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

Step 12.2

Multiply $x$ by $x^{3}$ .

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Step 12.2.1

Raise $x$ to the power of $1$ .

$x^{1} x^{3} \cdot 5$

Step 12.2.2

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

$x^{1 + 3} \cdot 5$

Step 12.3

Add $1$ and $3$ .

$x^{4} \cdot 5$

Step 13

Move $5$ to the left of $x^{4}$ .

$5 x^{4}$