Algebra Examples

$\sqrt[12]{\frac{81 x^{4} y^{12}}{16 c^{8} d^{4}}}$

Step 1

Rewrite $81 x^{4} y^{12}$ as ${(3 x y^{3})}^{4}$ .

$\sqrt[12]{\frac{{(3 x y^{3})}^{4}}{16 c^{8} d^{4}}}$

Step 2

Rewrite $16 c^{8} d^{4}$ as ${(2 c^{2} d)}^{4}$ .

$\sqrt[12]{\frac{{(3 x y^{3})}^{4}}{{(2 c^{2} d)}^{4}}}$

Step 3

Rewrite $\frac{{(3 x y^{3})}^{4}}{{(2 c^{2} d)}^{4}}$ as ${(\frac{3 x y^{3}}{2 c^{2} d})}^{4}$ .

$\sqrt[12]{{(\frac{3 x y^{3}}{2 c^{2} d})}^{4}}$

Step 4

Rewrite $\sqrt[12]{{(\frac{3 x y^{3}}{2 c^{2} d})}^{4}}$ as $\sqrt[3]{\sqrt[4]{{(\frac{3 x y^{3}}{2 c^{2} d})}^{4}}}$ .

$\sqrt[3]{\sqrt[4]{{(\frac{3 x y^{3}}{2 c^{2} d})}^{4}}}$

Step 5

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt[3]{\frac{3 x y^{3}}{2 c^{2} d}}$

Step 6

Rewrite $\sqrt[3]{\frac{3 x y^{3}}{2 c^{2} d}}$ as $\frac{\sqrt[3]{3 x y^{3}}}{\sqrt[3]{2 c^{2} d}}$ .

$\frac{\sqrt[3]{3 x y^{3}}}{\sqrt[3]{2 c^{2} d}}$

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $3 x y^{3}$ as $y^{3} \cdot (3 x)$ .

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Step 7.1.1

Move $x$ .

$\frac{\sqrt[3]{3 y^{3} x}}{\sqrt[3]{2 c^{2} d}}$

Step 7.1.2

Reorder $3$ and $y^{3}$ .

$\frac{\sqrt[3]{y^{3} \cdot 3 x}}{\sqrt[3]{2 c^{2} d}}$

Step 7.1.3

Add parentheses.

$\frac{\sqrt[3]{y^{3} \cdot (3 x)}}{\sqrt[3]{2 c^{2} d}}$

Step 7.2

Pull terms out from under the radical.

$\frac{y \sqrt[3]{3 x}}{\sqrt[3]{2 c^{2} d}}$

Step 8

Multiply $\frac{y \sqrt[3]{3 x}}{\sqrt[3]{2 c^{2} d}}$ by $\frac{{\sqrt[3]{2 c^{2} d}}^{2}}{{\sqrt[3]{2 c^{2} d}}^{2}}$ .

$\frac{y \sqrt[3]{3 x}}{\sqrt[3]{2 c^{2} d}} \cdot \frac{{\sqrt[3]{2 c^{2} d}}^{2}}{{\sqrt[3]{2 c^{2} d}}^{2}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply $\frac{y \sqrt[3]{3 x}}{\sqrt[3]{2 c^{2} d}}$ by $\frac{{\sqrt[3]{2 c^{2} d}}^{2}}{{\sqrt[3]{2 c^{2} d}}^{2}}$ .

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{\sqrt[3]{2 c^{2} d} {\sqrt[3]{2 c^{2} d}}^{2}}$

Step 9.2

Raise $\sqrt[3]{2 c^{2} d}$ to the power of $1$ .

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{{\sqrt[3]{2 c^{2} d}}^{1} {\sqrt[3]{2 c^{2} d}}^{2}}$

Step 9.3

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

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{{\sqrt[3]{2 c^{2} d}}^{1 + 2}}$

Step 9.4

Add $1$ and $2$ .

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{{\sqrt[3]{2 c^{2} d}}^{3}}$

Step 9.5

Rewrite ${\sqrt[3]{2 c^{2} d}}^{3}$ as $2 c^{2} d$ .

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Step 9.5.1

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

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{{({(2 c^{2} d)}^{\frac{1}{3}})}^{3}}$

Step 9.5.2

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

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{{(2 c^{2} d)}^{\frac{1}{3} \cdot 3}}$

Step 9.5.3

Combine $\frac{1}{3}$ and $3$ .

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{{(2 c^{2} d)}^{\frac{3}{3}}}$

Step 9.5.4

Cancel the common factor of $3$ .

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Step 9.5.4.1

Cancel the common factor.

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{{(2 c^{2} d)}^{\frac{3}{3}}}$

Step 9.5.4.2

Rewrite the expression.

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{{(2 c^{2} d)}^{1}}$

Step 9.5.5

Simplify.

$\frac{y \sqrt[3]{3 x} {\sqrt[3]{2 c^{2} d}}^{2}}{2 c^{2} d}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite ${\sqrt[3]{2 c^{2} d}}^{2}$ as $\sqrt[3]{{(2 c^{2} d)}^{2}}$ .

$\frac{y \sqrt[3]{3 x} \sqrt[3]{{(2 c^{2} d)}^{2}}}{2 c^{2} d}$

Step 10.2

Apply the product rule to $2 c^{2} d$ .

$\frac{y \sqrt[3]{3 x} \sqrt[3]{{(2 c^{2})}^{2} d^{2}}}{2 c^{2} d}$

Step 10.3

Apply the product rule to $2 c^{2}$ .

$\frac{y \sqrt[3]{3 x} \sqrt[3]{2^{2} {(c^{2})}^{2} d^{2}}}{2 c^{2} d}$

Step 10.4

Raise $2$ to the power of $2$ .

$\frac{y \sqrt[3]{3 x} \sqrt[3]{4 {(c^{2})}^{2} d^{2}}}{2 c^{2} d}$

Step 10.5

Multiply the exponents in ${(c^{2})}^{2}$ .

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Step 10.5.1

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

$\frac{y \sqrt[3]{3 x} \sqrt[3]{4 c^{2 \cdot 2} d^{2}}}{2 c^{2} d}$

Step 10.5.2

Multiply $2$ by $2$ .

$\frac{y \sqrt[3]{3 x} \sqrt[3]{4 c^{4} d^{2}}}{2 c^{2} d}$

Step 10.6

Rewrite $4 c^{4} d^{2}$ as $c^{3} \cdot (4 c d^{2})$ .

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Step 10.6.1

Factor out $c^{3}$ .

$\frac{y \sqrt[3]{3 x} \sqrt[3]{4 (c^{3} c) d^{2}}}{2 c^{2} d}$

Step 10.6.2

Reorder $4$ and $c^{3}$ .

$\frac{y \sqrt[3]{3 x} \sqrt[3]{c^{3} \cdot 4 c d^{2}}}{2 c^{2} d}$

Step 10.6.3

Add parentheses.

$\frac{y \sqrt[3]{3 x} \sqrt[3]{c^{3} \cdot 4 (c d^{2})}}{2 c^{2} d}$

Step 10.6.4

Add parentheses.

$\frac{y \sqrt[3]{3 x} \sqrt[3]{c^{3} \cdot (4 c d^{2})}}{2 c^{2} d}$

Step 10.7

Pull terms out from under the radical.

$\frac{y \sqrt[3]{3 x} c \sqrt[3]{4 c d^{2}}}{2 c^{2} d}$

Step 10.8

Combine exponents.

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Step 10.8.1

Combine using the product rule for radicals.

$\frac{y c \sqrt[3]{3 x (4 c d^{2})}}{2 c^{2} d}$

Step 10.8.2

Multiply $4$ by $3$ .

$\frac{y c \sqrt[3]{12 x c d^{2}}}{2 c^{2} d}$

Step 11

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

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Step 11.1

Factor $c$ out of $y c \sqrt[3]{12 x c d^{2}}$ .

$\frac{c (y \sqrt[3]{12 x c d^{2}})}{2 c^{2} d}$

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $c$ out of $2 c^{2} d$ .

$\frac{c (y \sqrt[3]{12 x c d^{2}})}{c (2 c d)}$

Step 11.2.2

Cancel the common factor.

$\frac{c (y \sqrt[3]{12 x c d^{2}})}{c (2 c d)}$

Step 11.2.3

Rewrite the expression.

$\frac{y \sqrt[3]{12 x c d^{2}}}{2 c d}$