Simplify square root of (18x^5y^4)/(49xz^3)

$\sqrt{\frac{18 x^{5} y^{4}}{49 x z^{3}}}$

Step 1

Reduce the expression $\frac{18 x^{5} y^{4}}{49 x z^{3}}$ by cancelling the common factors.

Factor $x$ out of $18 x^{5} y^{4}$ .

$\sqrt{\frac{x (18 x^{4} y^{4})}{49 x z^{3}}}$

Factor $x$ out of $49 x z^{3}$ .

$\sqrt{\frac{x (18 x^{4} y^{4})}{x (49 z^{3})}}$

Cancel the common factor.

$\sqrt{\frac{x (18 x^{4} y^{4})}{x (49 z^{3})}}$

Rewrite the expression.

$\sqrt{\frac{18 x^{4} y^{4}}{49 z^{3}}}$

Step 2

Rewrite $\frac{18 x^{4} y^{4}}{49 z^{3}}$ as ${(\frac{3 x^{2} y^{2}}{7 z})}^{2} \frac{2}{z}$ .

Factor the perfect power ${(3 x^{2} y^{2})}^{2}$ out of $18 x^{4} y^{4}$ .

$\sqrt{\frac{{(3 x^{2} y^{2})}^{2} \cdot 2}{49 z^{3}}}$

Factor the perfect power ${(7 z)}^{2}$ out of $49 z^{3}$ .

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

Rearrange the fraction $\frac{{(3 x^{2} y^{2})}^{2} \cdot 2}{{(7 z)}^{2} z}$ .

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

Step 3

Pull terms out from under the radical.

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

Step 4

Rewrite $\sqrt{\frac{2}{z}}$ as $\frac{\sqrt{2}}{\sqrt{z}}$ .

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

Step 5

Combine.

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

Step 6

Multiply $\frac{3 x^{2} y^{2} \sqrt{2}}{7 z \sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

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

Step 7

Combine and simplify the denominator.

Multiply $\frac{3 x^{2} y^{2} \sqrt{2}}{7 z \sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

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

Move $\sqrt{z}$ .

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

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

$\frac{3 x^{2} y^{2} \sqrt{2} \sqrt{z}}{7 z ({\sqrt{z}}^{1} \sqrt{z})}$

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

$\frac{3 x^{2} y^{2} \sqrt{2} \sqrt{z}}{7 z ({\sqrt{z}}^{1} {\sqrt{z}}^{1})}$

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

$\frac{3 x^{2} y^{2} \sqrt{2} \sqrt{z}}{7 z {\sqrt{z}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

$\frac{3 x^{2} y^{2} \sqrt{2} \sqrt{z}}{7 z {(z^{\frac{1}{2}})}^{2}}$

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

$\frac{3 x^{2} y^{2} \sqrt{2} \sqrt{z}}{7 z \cdot z^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{3 x^{2} y^{2} \sqrt{2} \sqrt{z}}{7 z \cdot z^{1}}$

Simplify.

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

Step 8

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

Move $z$ .

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

Multiply $z$ by $z$ .

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

Step 9

Combine using the product rule for radicals.

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

Step 10

Move $2$ to the left of $z$ .

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

Simplify square root of (18x^5y^4)/(49xz^3)

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