Precalculus Examples

$\frac{{(x y^{3} z^{3})}^{2}}{{(x^{2} y^{4} z)}^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(x y^{3})}^{2} {(z^{3})}^{2}}{{(x^{2} y^{4} z)}^{3}}$

Step 1.2

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

$\frac{x^{2} {(y^{3})}^{2} {(z^{3})}^{2}}{{(x^{2} y^{4} z)}^{3}}$

Step 1.3

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

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Step 1.3.1

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

$\frac{x^{2} y^{3 \cdot 2} {(z^{3})}^{2}}{{(x^{2} y^{4} z)}^{3}}$

Step 1.3.2

Multiply $3$ by $2$ .

$\frac{x^{2} y^{6} {(z^{3})}^{2}}{{(x^{2} y^{4} z)}^{3}}$

Step 1.4

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

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Step 1.4.1

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

$\frac{x^{2} y^{6} z^{3 \cdot 2}}{{(x^{2} y^{4} z)}^{3}}$

Step 1.4.2

Multiply $3$ by $2$ .

$\frac{x^{2} y^{6} z^{6}}{{(x^{2} y^{4} z)}^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $x^{2} y^{4} z$ .

$\frac{x^{2} y^{6} z^{6}}{{(x^{2} y^{4})}^{3} z^{3}}$

Step 2.2

Apply the product rule to $x^{2} y^{4}$ .

$\frac{x^{2} y^{6} z^{6}}{{(x^{2})}^{3} {(y^{4})}^{3} z^{3}}$

Step 2.3

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

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Step 2.3.1

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

$\frac{x^{2} y^{6} z^{6}}{x^{2 \cdot 3} {(y^{4})}^{3} z^{3}}$

Step 2.3.2

Multiply $2$ by $3$ .

$\frac{x^{2} y^{6} z^{6}}{x^{6} {(y^{4})}^{3} z^{3}}$

Step 2.4

Multiply the exponents in ${(y^{4})}^{3}$ .

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Step 2.4.1

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

$\frac{x^{2} y^{6} z^{6}}{x^{6} y^{4 \cdot 3} z^{3}}$

Step 2.4.2

Multiply $4$ by $3$ .

$\frac{x^{2} y^{6} z^{6}}{x^{6} y^{12} z^{3}}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $x^{2}$ and $x^{6}$ .

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Step 3.1.1

Factor $x^{2}$ out of $x^{2} y^{6} z^{6}$ .

$\frac{x^{2} (y^{6} z^{6})}{x^{6} y^{12} z^{3}}$

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

Factor $x^{2}$ out of $x^{6} y^{12} z^{3}$ .

$\frac{x^{2} (y^{6} z^{6})}{x^{2} (x^{4} y^{12} z^{3})}$

Step 3.1.2.2

Cancel the common factor.

$\frac{x^{2} (y^{6} z^{6})}{x^{2} (x^{4} y^{12} z^{3})}$

Step 3.1.2.3

Rewrite the expression.

$\frac{y^{6} z^{6}}{x^{4} y^{12} z^{3}}$

Step 3.2

Cancel the common factor of $y^{6}$ and $y^{12}$ .

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Step 3.2.1

Factor $y^{6}$ out of $y^{6} z^{6}$ .

$\frac{y^{6} (z^{6})}{x^{4} y^{12} z^{3}}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $y^{6}$ out of $x^{4} y^{12} z^{3}$ .

$\frac{y^{6} (z^{6})}{y^{6} (x^{4} y^{6} z^{3})}$

Step 3.2.2.2

Cancel the common factor.

$\frac{y^{6} z^{6}}{y^{6} (x^{4} y^{6} z^{3})}$

Step 3.2.2.3

Rewrite the expression.

$\frac{z^{6}}{x^{4} y^{6} z^{3}}$

Step 3.3

Cancel the common factor of $z^{6}$ and $z^{3}$ .

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Step 3.3.1

Factor $z^{3}$ out of $z^{6}$ .

$\frac{z^{3} z^{3}}{x^{4} y^{6} z^{3}}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $z^{3}$ out of $x^{4} y^{6} z^{3}$ .

$\frac{z^{3} z^{3}}{z^{3} (x^{4} y^{6})}$

Step 3.3.2.2

Cancel the common factor.

$\frac{z^{3} z^{3}}{z^{3} (x^{4} y^{6})}$

Step 3.3.2.3

Rewrite the expression.

$\frac{z^{3}}{x^{4} y^{6}}$