Simplify square root of (2g^3)/(5z)

$\sqrt{\frac{2 g^{3}}{5 z}}$

Step 1

Rewrite $\sqrt{\frac{2 g^{3}}{5 z}}$ as $\frac{\sqrt{2 g^{3}}}{\sqrt{5 z}}$ .

$\frac{\sqrt{2 g^{3}}}{\sqrt{5 z}}$

Step 2

Simplify the numerator.

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

Factor out $g^{2}$ .

$\frac{\sqrt{2 (g^{2} g)}}{\sqrt{5 z}}$

Reorder $2$ and $g^{2}$ .

$\frac{\sqrt{g^{2} \cdot 2 g}}{\sqrt{5 z}}$

Add parentheses.

$\frac{\sqrt{g^{2} \cdot (2 g)}}{\sqrt{5 z}}$

Pull terms out from under the radical.

$\frac{g \sqrt{2 g}}{\sqrt{5 z}}$

Step 3

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

$\frac{g \sqrt{2 g}}{\sqrt{5 z}} \cdot \frac{\sqrt{5 z}}{\sqrt{5 z}}$

Step 4

Combine and simplify the denominator.

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

$\frac{g \sqrt{2 g} \sqrt{5 z}}{\sqrt{5 z} \sqrt{5 z}}$

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

$\frac{g \sqrt{2 g} \sqrt{5 z}}{{\sqrt{5 z}}^{1} \sqrt{5 z}}$

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

$\frac{g \sqrt{2 g} \sqrt{5 z}}{{\sqrt{5 z}}^{1} {\sqrt{5 z}}^{1}}$

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

$\frac{g \sqrt{2 g} \sqrt{5 z}}{{\sqrt{5 z}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{g \sqrt{2 g} \sqrt{5 z}}{{\sqrt{5 z}}^{2}}$

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

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

$\frac{g \sqrt{2 g} \sqrt{5 z}}{{({(5 z)}^{\frac{1}{2}})}^{2}}$

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

$\frac{g \sqrt{2 g} \sqrt{5 z}}{{(5 z)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{g \sqrt{2 g} \sqrt{5 z}}{{(5 z)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{g \sqrt{2 g} \sqrt{5 z}}{{(5 z)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{g \sqrt{2 g} \sqrt{5 z}}{{(5 z)}^{1}}$

Simplify.

$\frac{g \sqrt{2 g} \sqrt{5 z}}{5 z}$

Step 5

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{g \sqrt{5 z (2 g)}}{5 z}$

Multiply $2$ by $5$ .

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

Simplify square root of (2g^3)/(5z)

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