Algebra Examples

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

Step 1

Multiply $\frac{\sqrt[4]{x^{2} y^{3}}}{\sqrt[3]{x y}}$ by $\frac{{\sqrt[3]{x y}}^{2}}{{\sqrt[3]{x y}}^{2}}$ .

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

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[4]{x^{2} y^{3}}}{\sqrt[3]{x y}}$ by $\frac{{\sqrt[3]{x y}}^{2}}{{\sqrt[3]{x y}}^{2}}$ .

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

Step 2.2

Raise $\sqrt[3]{x y}$ to the power of $1$ .

$\frac{\sqrt[4]{x^{2} y^{3}} {\sqrt[3]{x y}}^{2}}{{\sqrt[3]{x y}}^{1} {\sqrt[3]{x y}}^{2}}$

Step 2.3

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

$\frac{\sqrt[4]{x^{2} y^{3}} {\sqrt[3]{x y}}^{2}}{{\sqrt[3]{x y}}^{1 + 2}}$

Step 2.4

Add $1$ and $2$ .

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

Step 2.5

Rewrite ${\sqrt[3]{x y}}^{3}$ as $x y$ .

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Step 2.5.1

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

$\frac{\sqrt[4]{x^{2} y^{3}} {\sqrt[3]{x y}}^{2}}{{({(x y)}^{\frac{1}{3}})}^{3}}$

Step 2.5.2

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

$\frac{\sqrt[4]{x^{2} y^{3}} {\sqrt[3]{x y}}^{2}}{{(x y)}^{\frac{1}{3} \cdot 3}}$

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $3$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt[4]{x^{2} y^{3}} {\sqrt[3]{x y}}^{2}}{{(x y)}^{1}}$

Step 2.5.5

Simplify.

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

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{x y}}^{2}$ as $\sqrt[3]{{(x y)}^{2}}$ .

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

Step 3.2

Apply the product rule to $x y$ .

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

Step 4

Simplify the numerator.

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Step 4.1

Rewrite the expression using the least common index of $12$ .

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Step 4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{2} y^{3}}$ as ${(x^{2} y^{3})}^{\frac{1}{4}}$ .

$\frac{{(x^{2} y^{3})}^{\frac{1}{4}} \sqrt[3]{x^{2} y^{2}}}{x y}$

Step 4.1.2

Rewrite ${(x^{2} y^{3})}^{\frac{1}{4}}$ as ${(x^{2} y^{3})}^{\frac{3}{12}}$ .

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

Step 4.1.3

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

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

Step 4.1.4

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

$\frac{\sqrt[12]{{(x^{2} y^{3})}^{3}} {(x^{2} y^{2})}^{\frac{1}{3}}}{x y}$

Step 4.1.5

Rewrite ${(x^{2} y^{2})}^{\frac{1}{3}}$ as ${(x^{2} y^{2})}^{\frac{4}{12}}$ .

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

Step 4.1.6

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

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

Step 4.2

Combine using the product rule for radicals.

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

Step 4.3

Apply the product rule to $x^{2} y^{3}$ .

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

Step 4.4

Apply the product rule to $x^{2} y^{2}$ .

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

Step 4.5

Multiply ${(x^{2})}^{3}$ by ${(x^{2})}^{4}$ by adding the exponents.

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Step 4.5.1

Move ${(x^{2})}^{4}$ .

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

Step 4.5.2

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

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

Step 4.5.3

Add $4$ and $3$ .

$\frac{\sqrt[12]{{(x^{2})}^{7} {(y^{3})}^{3} {(y^{2})}^{4}}}{x y}$

Step 4.6

Multiply the exponents in ${(x^{2})}^{7}$ .

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Step 4.6.1

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

$\frac{\sqrt[12]{x^{2 \cdot 7} {(y^{3})}^{3} {(y^{2})}^{4}}}{x y}$

Step 4.6.2

Multiply $2$ by $7$ .

$\frac{\sqrt[12]{x^{14} {(y^{3})}^{3} {(y^{2})}^{4}}}{x y}$

Step 4.7

Multiply the exponents in ${(y^{3})}^{3}$ .

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Step 4.7.1

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

$\frac{\sqrt[12]{x^{14} y^{3 \cdot 3} {(y^{2})}^{4}}}{x y}$

Step 4.7.2

Multiply $3$ by $3$ .

$\frac{\sqrt[12]{x^{14} y^{9} {(y^{2})}^{4}}}{x y}$

Step 4.8

Multiply the exponents in ${(y^{2})}^{4}$ .

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Step 4.8.1

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

$\frac{\sqrt[12]{x^{14} y^{9} y^{2 \cdot 4}}}{x y}$

Step 4.8.2

Multiply $2$ by $4$ .

$\frac{\sqrt[12]{x^{14} y^{9} y^{8}}}{x y}$

Step 4.9

Multiply $y^{9}$ by $y^{8}$ by adding the exponents.

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Step 4.9.1

Move $y^{8}$ .

$\frac{\sqrt[12]{x^{14} (y^{8} y^{9})}}{x y}$

Step 4.9.2

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

$\frac{\sqrt[12]{x^{14} y^{8 + 9}}}{x y}$

Step 4.9.3

Add $8$ and $9$ .

$\frac{\sqrt[12]{x^{14} y^{17}}}{x y}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $x^{14} y^{17}$ as ${(x y)}^{12} (x^{2} y^{5})$ .

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Step 5.1.1

Factor out $x^{12}$ .

$\frac{\sqrt[12]{x^{12} x^{2} y^{17}}}{x y}$

Step 5.1.2

Factor out $y^{12}$ .

$\frac{\sqrt[12]{x^{12} x^{2} (y^{12} y^{5})}}{x y}$

Step 5.1.3

Move $x^{2}$ .

$\frac{\sqrt[12]{x^{12} y^{12} x^{2} y^{5}}}{x y}$

Step 5.1.4

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

$\frac{\sqrt[12]{{(x y)}^{12} x^{2} y^{5}}}{x y}$

Step 5.1.5

Add parentheses.

$\frac{\sqrt[12]{{(x y)}^{12} (x^{2} y^{5})}}{x y}$

Step 5.2

Pull terms out from under the radical.

$\frac{x y \sqrt[12]{x^{2} y^{5}}}{x y}$

Step 6

Cancel the common factor of $x$ .

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Step 6.1

Cancel the common factor.

$\frac{x y \sqrt[12]{x^{2} y^{5}}}{x y}$

Step 6.2

Rewrite the expression.

$\frac{y \sqrt[12]{x^{2} y^{5}}}{y}$

Step 7

Cancel the common factor of $y$ .

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Step 7.1

Cancel the common factor.

$\frac{y \sqrt[12]{x^{2} y^{5}}}{y}$

Step 7.2

Divide $\sqrt[12]{x^{2} y^{5}}$ by $1$ .

$\sqrt[12]{x^{2} y^{5}}$