Simplify square root of (26x^3y)/(47z^5r^3)

$\sqrt{\frac{26 x^{3} y}{47 z^{5} r^{3}}}$

Step 1

Rearrange the fraction $\frac{x^{2} \cdot (26 x y)}{{(z^{2} r)}^{2} \cdot (47 z r)}$ .

$\sqrt{{(\frac{x}{z^{2} r})}^{2} \frac{26 x y}{47 z r}}$

Step 2

Pull terms out from under the radical.

$\frac{x}{z^{2} r} \sqrt{\frac{26 x y}{47 z r}}$

Step 3

Rewrite $\sqrt{\frac{26 x y}{47 z r}}$ as $\frac{\sqrt{26 x y}}{\sqrt{47 z r}}$ .

$\frac{x}{z^{2} r} \cdot \frac{\sqrt{26 x y}}{\sqrt{47 z r}}$

Step 4

Combine.

$\frac{x \sqrt{26 x y}}{z^{2} r \sqrt{47 z r}}$

Step 5

Multiply $\frac{x \sqrt{26 x y}}{z^{2} r \sqrt{47 z r}}$ by $\frac{\sqrt{47 z r}}{\sqrt{47 z r}}$ .

$\frac{x \sqrt{26 x y}}{z^{2} r \sqrt{47 z r}} \cdot \frac{\sqrt{47 z r}}{\sqrt{47 z r}}$

Step 6

Combine and simplify the denominator.

Multiply $\frac{x \sqrt{26 x y}}{z^{2} r \sqrt{47 z r}}$ by $\frac{\sqrt{47 z r}}{\sqrt{47 z r}}$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r \sqrt{47 z r} \sqrt{47 z r}}$

Move $\sqrt{47 z r}$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r (\sqrt{47 z r} \sqrt{47 z r})}$

Raise $\sqrt{47 z r}$ to the power of $1$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r ({\sqrt{47 z r}}^{1} \sqrt{47 z r})}$

Raise $\sqrt{47 z r}$ to the power of $1$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r ({\sqrt{47 z r}}^{1} {\sqrt{47 z r}}^{1})}$

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

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r {\sqrt{47 z r}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r {\sqrt{47 z r}}^{2}}$

Rewrite ${\sqrt{47 z r}}^{2}$ as $47 z r$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{47 z r}$ as ${(47 z r)}^{\frac{1}{2}}$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r {({(47 z r)}^{\frac{1}{2}})}^{2}}$

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

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r {(47 z r)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r {(47 z r)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r {(47 z r)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r {(47 z r)}^{1}}$

Simplify.

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{2} r (47 z r)}$

Step 7

Multiply $z^{2}$ by $z$ by adding the exponents.

Move $z$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z \cdot z^{2} r (47 r)}$

Multiply $z$ by $z^{2}$ .

Raise $z$ to the power of $1$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{1} z^{2} r (47 r)}$

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

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{1 + 2} r (47 r)}$

Add $1$ and $2$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{3} r (47 r)}$

Step 8

Multiply $r$ by $r$ by adding the exponents.

Move $r$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{3} (r \cdot r) \cdot 47}$

Multiply $r$ by $r$ .

$\frac{x \sqrt{26 x y} \sqrt{47 z r}}{z^{3} r^{2} \cdot 47}$

Step 9

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{x \sqrt{47 z r (26 x y)}}{z^{3} r^{2} \cdot 47}$

Multiply $26$ by $47$ .

$\frac{x \sqrt{1222 z r x y}}{z^{3} r^{2} \cdot 47}$

Step 10

Move $47$ to the left of $z^{3} r^{2}$ .

$\frac{x \sqrt{1222 z r x y}}{47 z^{3} r^{2}}$

Simplify square root of (26x^3y)/(47z^5r^3)

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