Simplify square root of (8x^5y^7z^4)/(24x^3y^2z^7)

$\sqrt{\frac{8 x^{5} y^{7} z^{4}}{24 x^{3} y^{2} z^{7}}}$

Step 1

Reduce the expression $\frac{8 x^{5} y^{7} z^{4}}{24 x^{3} y^{2} z^{7}}$ by cancelling the common factors.

Factor $8$ out of $8 x^{5} y^{7} z^{4}$ .

$\sqrt{\frac{8 (x^{5} y^{7} z^{4})}{24 x^{3} y^{2} z^{7}}}$

Factor $8$ out of $24 x^{3} y^{2} z^{7}$ .

$\sqrt{\frac{8 (x^{5} y^{7} z^{4})}{8 (3 x^{3} y^{2} z^{7})}}$

Cancel the common factor.

$\sqrt{\frac{8 (x^{5} y^{7} z^{4})}{8 (3 x^{3} y^{2} z^{7})}}$

Rewrite the expression.

$\sqrt{\frac{x^{5} y^{7} z^{4}}{3 x^{3} y^{2} z^{7}}}$

Step 2

Cancel the common factor of $x^{5}$ and $x^{3}$ .

Factor $x^{3}$ out of $x^{5} y^{7} z^{4}$ .

$\sqrt{\frac{x^{3} (x^{2} y^{7} z^{4})}{3 x^{3} y^{2} z^{7}}}$

Cancel the common factors.

Factor $x^{3}$ out of $3 x^{3} y^{2} z^{7}$ .

$\sqrt{\frac{x^{3} (x^{2} y^{7} z^{4})}{x^{3} (3 y^{2} z^{7})}}$

Cancel the common factor.

$\sqrt{\frac{x^{3} (x^{2} y^{7} z^{4})}{x^{3} (3 y^{2} z^{7})}}$

Rewrite the expression.

$\sqrt{\frac{x^{2} y^{7} z^{4}}{3 y^{2} z^{7}}}$

Step 3

Cancel the common factor of $y^{7}$ and $y^{2}$ .

Factor $y^{2}$ out of $x^{2} y^{7} z^{4}$ .

$\sqrt{\frac{y^{2} (x^{2} y^{5} z^{4})}{3 y^{2} z^{7}}}$

Cancel the common factors.

Factor $y^{2}$ out of $3 y^{2} z^{7}$ .

$\sqrt{\frac{y^{2} (x^{2} y^{5} z^{4})}{y^{2} (3 z^{7})}}$

Cancel the common factor.

$\sqrt{\frac{y^{2} (x^{2} y^{5} z^{4})}{y^{2} (3 z^{7})}}$

Rewrite the expression.

$\sqrt{\frac{x^{2} y^{5} z^{4}}{3 z^{7}}}$

Step 4

Cancel the common factor of $z^{4}$ and $z^{7}$ .

Factor $z^{4}$ out of $x^{2} y^{5} z^{4}$ .

$\sqrt{\frac{z^{4} (x^{2} y^{5})}{3 z^{7}}}$

Cancel the common factors.

Factor $z^{4}$ out of $3 z^{7}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\sqrt{\frac{x^{2} y^{5}}{3 z^{3}}}$

Step 5

Rewrite $\frac{x^{2} y^{5}}{3 z^{3}}$ as ${(\frac{x y^{2}}{z})}^{2} \frac{y}{3 z}$ .

Factor the perfect power ${(x y^{2})}^{2}$ out of $x^{2} y^{5}$ .

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

Factor the perfect power $z^{2}$ out of $3 z^{3}$ .

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

Rearrange the fraction $\frac{{(x y^{2})}^{2} y}{z^{2} \cdot (3 z)}$ .

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

Step 6

Pull terms out from under the radical.

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

Step 7

Rewrite $\sqrt{\frac{y}{3 z}}$ as $\frac{\sqrt{y}}{\sqrt{3 z}}$ .

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

Step 8

Combine.

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

Step 9

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

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

Step 10

Combine and simplify the denominator.

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

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

Move $\sqrt{3 z}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Step 11

Multiply $z$ by $z$ by adding the exponents.

Move $z$ .

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

Multiply $z$ by $z$ .

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

Step 12

Combine using the product rule for radicals.

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

Step 13

Move $3$ to the left of $z^{2}$ .

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

Simplify square root of (8x^5y^7z^4)/(24x^3y^2z^7)

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