Algebra Examples

$\frac{\sqrt[5]{2 x y}}{\sqrt[4]{3 x^{2} y}}$

Step 1

Multiply $\frac{\sqrt[5]{2 x y}}{\sqrt[4]{3 x^{2} y}}$ by $\frac{{\sqrt[4]{3 x^{2} y}}^{3}}{{\sqrt[4]{3 x^{2} y}}^{3}}$ .

$\frac{\sqrt[5]{2 x y}}{\sqrt[4]{3 x^{2} y}} \cdot \frac{{\sqrt[4]{3 x^{2} y}}^{3}}{{\sqrt[4]{3 x^{2} y}}^{3}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{\sqrt[5]{2 x y}}{\sqrt[4]{3 x^{2} y}}$ by $\frac{{\sqrt[4]{3 x^{2} y}}^{3}}{{\sqrt[4]{3 x^{2} y}}^{3}}$ .

$\frac{\sqrt[5]{2 x y} {\sqrt[4]{3 x^{2} y}}^{3}}{\sqrt[4]{3 x^{2} y} {\sqrt[4]{3 x^{2} y}}^{3}}$

Step 2.2

Raise $\sqrt[4]{3 x^{2} y}$ to the power of $1$ .

$\frac{\sqrt[5]{2 x y} {\sqrt[4]{3 x^{2} y}}^{3}}{{\sqrt[4]{3 x^{2} y}}^{1} {\sqrt[4]{3 x^{2} y}}^{3}}$

Step 2.3

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

$\frac{\sqrt[5]{2 x y} {\sqrt[4]{3 x^{2} y}}^{3}}{{\sqrt[4]{3 x^{2} y}}^{1 + 3}}$

Step 2.4

Add $1$ and $3$ .

$\frac{\sqrt[5]{2 x y} {\sqrt[4]{3 x^{2} y}}^{3}}{{\sqrt[4]{3 x^{2} y}}^{4}}$

Step 2.5

Rewrite ${\sqrt[4]{3 x^{2} y}}^{4}$ as $3 x^{2} y$ .

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Step 2.5.1

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

$\frac{\sqrt[5]{2 x y} {\sqrt[4]{3 x^{2} y}}^{3}}{{({(3 x^{2} y)}^{\frac{1}{4}})}^{4}}$

Step 2.5.2

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

$\frac{\sqrt[5]{2 x y} {\sqrt[4]{3 x^{2} y}}^{3}}{{(3 x^{2} y)}^{\frac{1}{4} \cdot 4}}$

Step 2.5.3

Combine $\frac{1}{4}$ and $4$ .

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

Step 2.5.4

Cancel the common factor of $4$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

$\frac{\sqrt[5]{2 x y} {\sqrt[4]{3 x^{2} y}}^{3}}{{(3 x^{2} y)}^{1}}$

Step 2.5.5

Simplify.

$\frac{\sqrt[5]{2 x y} {\sqrt[4]{3 x^{2} y}}^{3}}{3 x^{2} y}$

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Raise $3$ to the power of $3$ .

$\frac{\sqrt[5]{2 x y} \sqrt[4]{27 {(x^{2})}^{3} y^{3}}}{3 x^{2} y}$

Step 3.5

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 3.5.1

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

$\frac{\sqrt[5]{2 x y} \sqrt[4]{27 x^{2 \cdot 3} y^{3}}}{3 x^{2} y}$

Step 3.5.2

Multiply $2$ by $3$ .

$\frac{\sqrt[5]{2 x y} \sqrt[4]{27 x^{6} y^{3}}}{3 x^{2} y}$

Step 3.6

Rewrite $27 x^{6} y^{3}$ as $x^{4} \cdot (27 x^{2} y^{3})$ .

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Step 3.6.1

Factor out $x^{4}$ .

$\frac{\sqrt[5]{2 x y} \sqrt[4]{27 (x^{4} x^{2}) y^{3}}}{3 x^{2} y}$

Step 3.6.2

Reorder $27$ and $x^{4}$ .

$\frac{\sqrt[5]{2 x y} \sqrt[4]{x^{4} \cdot 27 x^{2} y^{3}}}{3 x^{2} y}$

Step 3.6.3

Add parentheses.

$\frac{\sqrt[5]{2 x y} \sqrt[4]{x^{4} \cdot 27 (x^{2} y^{3})}}{3 x^{2} y}$

Step 3.6.4

Add parentheses.

$\frac{\sqrt[5]{2 x y} \sqrt[4]{x^{4} \cdot (27 x^{2} y^{3})}}{3 x^{2} y}$

Step 3.7

Pull terms out from under the radical.

$\frac{\sqrt[5]{2 x y} x \sqrt[4]{27 x^{2} y^{3}}}{3 x^{2} y}$

Step 3.8

Combine exponents.

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Step 3.8.1

Rewrite the expression using the least common index of $20$ .

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Step 3.8.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2 x y}$ as ${(2 x y)}^{\frac{1}{5}}$ .

$\frac{x ({(2 x y)}^{\frac{1}{5}} \sqrt[4]{27 x^{2} y^{3}})}{3 x^{2} y}$

Step 3.8.1.2

Rewrite ${(2 x y)}^{\frac{1}{5}}$ as ${(2 x y)}^{\frac{4}{20}}$ .

$\frac{x ({(2 x y)}^{\frac{4}{20}} \sqrt[4]{27 x^{2} y^{3}})}{3 x^{2} y}$

Step 3.8.1.3

Rewrite ${(2 x y)}^{\frac{4}{20}}$ as $\sqrt[20]{{(2 x y)}^{4}}$ .

$\frac{x (\sqrt[20]{{(2 x y)}^{4}} \sqrt[4]{27 x^{2} y^{3}})}{3 x^{2} y}$

Step 3.8.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{27 x^{2} y^{3}}$ as ${(27 x^{2} y^{3})}^{\frac{1}{4}}$ .

$\frac{x (\sqrt[20]{{(2 x y)}^{4}} {(27 x^{2} y^{3})}^{\frac{1}{4}})}{3 x^{2} y}$

Step 3.8.1.5

Rewrite ${(27 x^{2} y^{3})}^{\frac{1}{4}}$ as ${(27 x^{2} y^{3})}^{\frac{5}{20}}$ .

$\frac{x (\sqrt[20]{{(2 x y)}^{4}} {(27 x^{2} y^{3})}^{\frac{5}{20}})}{3 x^{2} y}$

Step 3.8.1.6

Rewrite ${(27 x^{2} y^{3})}^{\frac{5}{20}}$ as $\sqrt[20]{{(27 x^{2} y^{3})}^{5}}$ .

$\frac{x (\sqrt[20]{{(2 x y)}^{4}} \sqrt[20]{{(27 x^{2} y^{3})}^{5}})}{3 x^{2} y}$

Step 3.8.2

Combine using the product rule for radicals.

$\frac{x \sqrt[20]{{(2 x y)}^{4} {(27 x^{2} y^{3})}^{5}}}{3 x^{2} y}$

Step 3.8.3

Apply the product rule to $2 x y$ .

$\frac{x \sqrt[20]{{(2 x)}^{4} y^{4} {(27 x^{2} y^{3})}^{5}}}{3 x^{2} y}$

Step 3.8.4

Apply the product rule to $27 x^{2} y^{3}$ .

$\frac{x \sqrt[20]{{(2 x)}^{4} y^{4} {(27 x^{2})}^{5} {(y^{3})}^{5}}}{3 x^{2} y}$

Step 3.8.5

Apply the product rule to $2 x$ .

$\frac{x \sqrt[20]{2^{4} x^{4} y^{4} {(27 x^{2})}^{5} {(y^{3})}^{5}}}{3 x^{2} y}$

Step 3.8.6

Raise $2$ to the power of $4$ .

$\frac{x \sqrt[20]{16 x^{4} y^{4} {(27 x^{2})}^{5} {(y^{3})}^{5}}}{3 x^{2} y}$

Step 3.8.7

Apply the product rule to $27 x^{2}$ .

$\frac{x \sqrt[20]{16 x^{4} y^{4} (27^{5} {(x^{2})}^{5}) {(y^{3})}^{5}}}{3 x^{2} y}$

Step 3.8.8

Raise $27$ to the power of $5$ .

$\frac{x \sqrt[20]{16 x^{4} y^{4} (14348907 {(x^{2})}^{5}) {(y^{3})}^{5}}}{3 x^{2} y}$

Step 3.8.9

Multiply the exponents in ${(x^{2})}^{5}$ .

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Step 3.8.9.1

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

$\frac{x \sqrt[20]{16 x^{4} y^{4} (14348907 x^{2 \cdot 5}) {(y^{3})}^{5}}}{3 x^{2} y}$

Step 3.8.9.2

Multiply $2$ by $5$ .

$\frac{x \sqrt[20]{16 x^{4} y^{4} (14348907 x^{10}) {(y^{3})}^{5}}}{3 x^{2} y}$

Step 3.8.10

Multiply the exponents in ${(y^{3})}^{5}$ .

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Step 3.8.10.1

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

$\frac{x \sqrt[20]{16 x^{4} y^{4} (14348907 x^{10}) y^{3 \cdot 5}}}{3 x^{2} y}$

Step 3.8.10.2

Multiply $3$ by $5$ .

$\frac{x \sqrt[20]{16 x^{4} y^{4} (14348907 x^{10}) y^{15}}}{3 x^{2} y}$

$\frac{x \sqrt[20]{16 x^{4} y^{4} \cdot 14348907 x^{10} y^{15}}}{3 x^{2} y}$

Step 3.8.11

Combine exponents.

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Step 3.8.11.1

Multiply $14348907$ by $16$ .

$\frac{x \sqrt[20]{229582512 x^{4} y^{4} x^{10} y^{15}}}{3 x^{2} y}$

Step 3.8.11.2

Multiply $x^{4}$ by $x^{10}$ by adding the exponents.

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Step 3.8.11.2.1

Move $x^{10}$ .

$\frac{x \sqrt[20]{229582512 (x^{10} x^{4}) y^{4} y^{15}}}{3 x^{2} y}$

Step 3.8.11.2.2

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

$\frac{x \sqrt[20]{229582512 x^{10 + 4} y^{4} y^{15}}}{3 x^{2} y}$

Step 3.8.11.2.3

Add $10$ and $4$ .

$\frac{x \sqrt[20]{229582512 x^{14} y^{4} y^{15}}}{3 x^{2} y}$

Step 3.8.11.3

Multiply $y^{4}$ by $y^{15}$ by adding the exponents.

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Step 3.8.11.3.1

Move $y^{15}$ .

$\frac{x \sqrt[20]{229582512 x^{14} (y^{15} y^{4})}}{3 x^{2} y}$

Step 3.8.11.3.2

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

$\frac{x \sqrt[20]{229582512 x^{14} y^{15 + 4}}}{3 x^{2} y}$

Step 3.8.11.3.3

Add $15$ and $4$ .

$\frac{x \sqrt[20]{229582512 x^{14} y^{19}}}{3 x^{2} y}$

Step 4

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

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Step 4.1

Factor $x$ out of $x \sqrt[20]{229582512 x^{14} y^{19}}$ .

$\frac{x (\sqrt[20]{229582512 x^{14} y^{19}})}{3 x^{2} y}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\frac{x (\sqrt[20]{229582512 x^{14} y^{19}})}{x (3 x y)}$

Step 4.2.2

Cancel the common factor.

$\frac{x \sqrt[20]{229582512 x^{14} y^{19}}}{x (3 x y)}$

Step 4.2.3

Rewrite the expression.

$\frac{\sqrt[20]{229582512 x^{14} y^{19}}}{3 x y}$