Simplify ( square root of 10x^4)/( square root of 15x^3)

$\frac{\sqrt{10 x^{4}}}{\sqrt{15 x^{3}}}$

Step 1

Combine $\sqrt{10 x^{4}}$ and $\sqrt{15 x^{3}}$ into a single radical.

$\sqrt{\frac{10 x^{4}}{15 x^{3}}}$

Step 2

Reduce the expression $\frac{10 x^{4}}{15 x^{3}}$ by cancelling the common factors.

Factor $5$ out of $10 x^{4}$ .

$\sqrt{\frac{5 (2 x^{4})}{15 x^{3}}}$

Factor $5$ out of $15 x^{3}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{\frac{2 x^{4}}{3 x^{3}}}$

Step 3

Cancel the common factor of $x^{4}$ and $x^{3}$ .

Factor $x^{3}$ out of $2 x^{4}$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

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

Step 5

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

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

Step 6

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 7

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

$\frac{\sqrt{6 x}}{3}$

Simplify ( square root of 10x^4)/( square root of 15x^3)

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