Algebra Examples

$\sqrt[3]{\frac{\sqrt{x^{6} y^{8} z}}{125 z^{\frac{1}{2}}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $x^{6} y^{8} z$ as ${(x^{3} y^{4})}^{2} z$ .

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Step 1.1.1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$\sqrt[3]{\frac{\sqrt{{(x^{3})}^{2} y^{8} z}}{125 z^{\frac{1}{2}}}}$

Step 1.1.2

Rewrite $y^{8}$ as ${(y^{4})}^{2}$ .

$\sqrt[3]{\frac{\sqrt{{(x^{3})}^{2} {(y^{4})}^{2} z}}{125 z^{\frac{1}{2}}}}$

Step 1.1.3

Rewrite ${(x^{3})}^{2} {(y^{4})}^{2}$ as ${(x^{3} y^{4})}^{2}$ .

$\sqrt[3]{\frac{\sqrt{{(x^{3} y^{4})}^{2} z}}{125 z^{\frac{1}{2}}}}$

Step 1.2

Pull terms out from under the radical.

$\sqrt[3]{\frac{x^{3} y^{4} \sqrt{z}}{125 z^{\frac{1}{2}}}}$

Step 2

Rewrite $\frac{x^{3} y^{4} \sqrt{z}}{125 z^{\frac{1}{2}}}$ as ${(\frac{x y}{5})}^{3} \frac{y \sqrt{z}}{z^{\frac{1^{3}}{2}}}$ .

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Step 2.1

Factor the perfect power ${(x y)}^{3}$ out of $x^{3} y^{4} \sqrt{z}$ .

$\sqrt[3]{\frac{{(x y)}^{3} (y \sqrt{z})}{125 z^{\frac{1}{2}}}}$

Step 2.2

Factor the perfect power $5^{3}$ out of $125 z^{\frac{1}{2}}$ .

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

Step 2.3

Rearrange the fraction $\frac{{(x y)}^{3} (y \sqrt{z})}{5^{3} z^{\frac{1^{3}}{2}}}$ .

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

Step 3

Pull terms out from under the radical.

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

Step 4

One to any power is one.

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

Step 5

Rewrite $\sqrt[3]{\frac{y \sqrt{z}}{z^{\frac{1}{2}}}}$ as $\frac{\sqrt[3]{y \sqrt{z}}}{\sqrt[3]{z^{\frac{1}{2}}}}$ .

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

Step 6

Combine.

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

Step 7

Multiply $\frac{x y \sqrt[3]{y \sqrt{z}}}{5 \sqrt[3]{z^{\frac{1}{2}}}}$ by $\frac{{\sqrt[3]{z^{\frac{1}{2}}}}^{2}}{{\sqrt[3]{z^{\frac{1}{2}}}}^{2}}$ .

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

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{x y \sqrt[3]{y \sqrt{z}}}{5 \sqrt[3]{z^{\frac{1}{2}}}}$ by $\frac{{\sqrt[3]{z^{\frac{1}{2}}}}^{2}}{{\sqrt[3]{z^{\frac{1}{2}}}}^{2}}$ .

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

Step 8.2

Move $\sqrt[3]{z^{\frac{1}{2}}}$ .

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $2$ .

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

Step 8.6

Rewrite ${\sqrt[3]{z^{\frac{1}{2}}}}^{3}$ as $\sqrt{z}$ .

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Step 8.6.1

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

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

Step 8.6.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.6.3

Multiply $\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 8.6.3.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

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

Step 8.6.3.2

Multiply $2$ by $3$ .

$\frac{x y \sqrt[3]{y \sqrt{z}} {\sqrt[3]{z^{\frac{1}{2}}}}^{2}}{5 {(z^{\frac{1}{6}})}^{3}}$

Step 8.6.4

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

$\frac{x y \sqrt[3]{y \sqrt{z}} {\sqrt[3]{z^{\frac{1}{2}}}}^{2}}{5 z^{\frac{1}{6} \cdot 3}}$

Step 8.6.5

Combine $\frac{1}{6}$ and $3$ .

$\frac{x y \sqrt[3]{y \sqrt{z}} {\sqrt[3]{z^{\frac{1}{2}}}}^{2}}{5 z^{\frac{3}{6}}}$

Step 8.6.6

Cancel the common factor of $3$ and $6$ .

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Step 8.6.6.1

Factor $3$ out of $3$ .

$\frac{x y \sqrt[3]{y \sqrt{z}} {\sqrt[3]{z^{\frac{1}{2}}}}^{2}}{5 z^{\frac{3 (1)}{6}}}$

Step 8.6.6.2

Cancel the common factors.

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Step 8.6.6.2.1

Factor $3$ out of $6$ .

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

Step 8.6.6.2.2

Cancel the common factor.

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

Step 8.6.6.2.3

Rewrite the expression.

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

Step 8.6.7

Rewrite $z^{\frac{1}{2}}$ as $\sqrt{z}$ .

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

Step 9

Simplify the numerator.

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Step 9.1

Rewrite ${\sqrt[3]{z^{\frac{1}{2}}}}^{2}$ as $\sqrt[3]{z}$ .

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Step 9.1.1

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

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

Step 9.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.1.3

Multiply $\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 9.1.3.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

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

Step 9.1.3.2

Multiply $2$ by $3$ .

$\frac{x y \sqrt[3]{y \sqrt{z}} {(z^{\frac{1}{6}})}^{2}}{5 \sqrt{z}}$

Step 9.1.4

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

$\frac{x y \sqrt[3]{y \sqrt{z}} z^{\frac{1}{6} \cdot 2}}{5 \sqrt{z}}$

Step 9.1.5

Combine $\frac{1}{6}$ and $2$ .

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

Step 9.1.6

Cancel the common factor of $2$ and $6$ .

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Step 9.1.6.1

Factor $2$ out of $2$ .

$\frac{x y \sqrt[3]{y \sqrt{z}} z^{\frac{2 (1)}{6}}}{5 \sqrt{z}}$

Step 9.1.6.2

Cancel the common factors.

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Step 9.1.6.2.1

Factor $2$ out of $6$ .

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

Step 9.1.6.2.2

Cancel the common factor.

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

Step 9.1.6.2.3

Rewrite the expression.

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

Step 9.1.7

Rewrite $z^{\frac{1}{3}}$ as $\sqrt[3]{z}$ .

$\frac{x y \sqrt[3]{y \sqrt{z}} \sqrt[3]{z}}{5 \sqrt{z}}$

Step 9.2

Combine using the product rule for radicals.

$\frac{x y \sqrt[3]{z y \sqrt{z}}}{5 \sqrt{z}}$

Step 10

Multiply $\frac{x y \sqrt[3]{z y \sqrt{z}}}{5 \sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$\frac{x y \sqrt[3]{z y \sqrt{z}}}{5 \sqrt{z}} \cdot \frac{\sqrt{z}}{\sqrt{z}}$

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{x y \sqrt[3]{z y \sqrt{z}}}{5 \sqrt{z}}$ by $\frac{\sqrt{z}}{\sqrt{z}}$ .

$\frac{x y \sqrt[3]{z y \sqrt{z}} \sqrt{z}}{5 \sqrt{z} \sqrt{z}}$

Step 11.2

Move $\sqrt{z}$ .

$\frac{x y \sqrt[3]{z y \sqrt{z}} \sqrt{z}}{5 (\sqrt{z} \sqrt{z})}$

Step 11.3

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

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

Step 11.4

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

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

Step 11.5

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

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

Step 11.6

Add $1$ and $1$ .

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

Step 11.7

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

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Step 11.7.1

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

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

Step 11.7.2

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

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

Step 11.7.3

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

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

Step 11.7.4

Cancel the common factor of $2$ .

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Step 11.7.4.1

Cancel the common factor.

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

Step 11.7.4.2

Rewrite the expression.

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

Step 11.7.5

Simplify.

$\frac{x y \sqrt[3]{z y \sqrt{z}} \sqrt{z}}{5 z}$

Step 12

Simplify the numerator.

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Step 12.1

Rewrite the expression using the least common index of $6$ .

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Step 12.1.1

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

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

Step 12.1.2

Rewrite $z^{\frac{1}{2}}$ as $z^{\frac{3}{6}}$ .

$\frac{x y (z^{\frac{3}{6}} \sqrt[3]{z y \sqrt{z}})}{5 z}$

Step 12.1.3

Rewrite $z^{\frac{3}{6}}$ as $\sqrt[6]{z^{3}}$ .

$\frac{x y (\sqrt[6]{z^{3}} \sqrt[3]{z y \sqrt{z}})}{5 z}$

Step 12.1.4

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

$\frac{x y (\sqrt[6]{z^{3}} {(z y \sqrt{z})}^{\frac{1}{3}})}{5 z}$

Step 12.1.5

Rewrite ${(z y \sqrt{z})}^{\frac{1}{3}}$ as ${(z y \sqrt{z})}^{\frac{2}{6}}$ .

$\frac{x y (\sqrt[6]{z^{3}} {(z y \sqrt{z})}^{\frac{2}{6}})}{5 z}$

Step 12.1.6

Rewrite ${(z y \sqrt{z})}^{\frac{2}{6}}$ as $\sqrt[6]{{(z y \sqrt{z})}^{2}}$ .

$\frac{x y (\sqrt[6]{z^{3}} \sqrt[6]{{(z y \sqrt{z})}^{2}})}{5 z}$

Step 12.2

Combine using the product rule for radicals.

$\frac{x y \sqrt[6]{z^{3} {(z y \sqrt{z})}^{2}}}{5 z}$

Step 12.3

Apply the product rule to $z y \sqrt{z}$ .

$\frac{x y \sqrt[6]{z^{3} {(z y)}^{2} {\sqrt{z}}^{2}}}{5 z}$

Step 12.4

Apply the product rule to $z y$ .

$\frac{x y \sqrt[6]{z^{3} (z^{2} y^{2}) {\sqrt{z}}^{2}}}{5 z}$

Step 12.5

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

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Step 12.5.1

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

$\frac{x y \sqrt[6]{z^{3} (z^{2} y^{2}) {(z^{\frac{1}{2}})}^{2}}}{5 z}$

Step 12.5.2

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

$\frac{x y \sqrt[6]{z^{3} (z^{2} y^{2}) z^{\frac{1}{2} \cdot 2}}}{5 z}$

Step 12.5.3

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

$\frac{x y \sqrt[6]{z^{3} (z^{2} y^{2}) z^{\frac{2}{2}}}}{5 z}$

Step 12.5.4

Cancel the common factor of $2$ .

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Step 12.5.4.1

Cancel the common factor.

$\frac{x y \sqrt[6]{z^{3} (z^{2} y^{2}) z^{\frac{2}{2}}}}{5 z}$

Step 12.5.4.2

Rewrite the expression.

$\frac{x y \sqrt[6]{z^{3} (z^{2} y^{2}) z^{1}}}{5 z}$

Step 12.5.5

Simplify.

$\frac{x y \sqrt[6]{z^{3} (z^{2} y^{2}) z}}{5 z}$

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

Step 12.6

Combine exponents.

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Step 12.6.1

Multiply $z^{3}$ by $z^{2}$ by adding the exponents.

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Step 12.6.1.1

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

$\frac{x y \sqrt[6]{z^{3 + 2} y^{2} z}}{5 z}$

Step 12.6.1.2

Add $3$ and $2$ .

$\frac{x y \sqrt[6]{z^{5} y^{2} z}}{5 z}$

Step 12.6.2

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

$\frac{x y \sqrt[6]{y^{2} (z^{5} z^{1})}}{5 z}$

Step 12.6.3

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

$\frac{x y \sqrt[6]{y^{2} z^{5 + 1}}}{5 z}$

Step 12.6.4

Add $5$ and $1$ .

$\frac{x y \sqrt[6]{y^{2} z^{6}}}{5 z}$

Step 13

Simplify the numerator.

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Step 13.1

Reorder $y^{2}$ and $z^{6}$ .

$\frac{x y \sqrt[6]{z^{6} y^{2}}}{5 z}$

Step 13.2

Pull terms out from under the radical.

$\frac{x y (z \sqrt[6]{y^{2}})}{5 z}$

Step 13.3

Rewrite $\sqrt[6]{y^{2}}$ as $\sqrt[3]{\sqrt{y^{2}}}$ .

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

Step 13.4

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

$\frac{x y z \sqrt[3]{y}}{5 z}$

Step 14

Cancel the common factor of $z$ .

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Step 14.1

Cancel the common factor.

$\frac{x y z \sqrt[3]{y}}{5 z}$

Step 14.2

Rewrite the expression.

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