Simplify ( square root of 7x)/( square root of 10x^3)

$\frac{\sqrt{7 x}}{\sqrt{10 x^{3}}}$

Step 1

Combine $\sqrt{7 x}$ and $\sqrt{10 x^{3}}$ into a single radical.

$\sqrt{\frac{7 x}{10 x^{3}}}$

Step 2

Reduce the expression $\frac{7 x}{10 x^{3}}$ by cancelling the common factors.

Factor $x$ out of $7 x$ .

$\sqrt{\frac{x \cdot 7}{10 x^{3}}}$

Factor $x$ out of $10 x^{3}$ .

$\sqrt{\frac{x \cdot 7}{x (10 x^{2})}}$

Cancel the common factor.

$\sqrt{\frac{x \cdot 7}{x (10 x^{2})}}$

Rewrite the expression.

$\sqrt{\frac{7}{10 x^{2}}}$

Step 3

Rewrite $\frac{7}{10 x^{2}}$ as ${(\frac{1}{x})}^{2} \frac{7}{10}$ .

Factor the perfect power $1^{2}$ out of $7$ .

$\sqrt{\frac{1^{2} \cdot 7}{10 x^{2}}}$

Factor the perfect power $x^{2}$ out of $10 x^{2}$ .

$\sqrt{\frac{1^{2} \cdot 7}{x^{2} \cdot 10}}$

Rearrange the fraction $\frac{1^{2} \cdot 7}{x^{2} \cdot 10}$ .

$\sqrt{{(\frac{1}{x})}^{2} \frac{7}{10}}$

Step 4

Pull terms out from under the radical.

$\frac{1}{x} \sqrt{\frac{7}{10}}$

Step 5

Rewrite $\sqrt{\frac{7}{10}}$ as $\frac{\sqrt{7}}{\sqrt{10}}$ .

$\frac{1}{x} \cdot \frac{\sqrt{7}}{\sqrt{10}}$

Step 6

Multiply $\frac{\sqrt{7}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{1}{x} (\frac{\sqrt{7}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 7

Combine and simplify the denominator.

Multiply $\frac{\sqrt{7}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Raise $\sqrt{10}$ to the power of $1$ .

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

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

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{{\sqrt{10}}^{2}}$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

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

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

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

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{10^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{10^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{10^{1}}$

Evaluate the exponent.

$\frac{1}{x} \cdot \frac{\sqrt{7} \sqrt{10}}{10}$

Step 8

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{1}{x} \cdot \frac{\sqrt{7 \cdot 10}}{10}$

Multiply $7$ by $10$ .

$\frac{1}{x} \cdot \frac{\sqrt{70}}{10}$

Step 9

Combine fractions.

Multiply $\frac{1}{x}$ by $\frac{\sqrt{70}}{10}$ .

$\frac{\sqrt{70}}{x \cdot 10}$

Move $10$ to the left of $x$ .

$\frac{\sqrt{70}}{10 x}$

Simplify ( square root of 7x)/( square root of 10x^3)

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