Precalculus Examples

$(\frac{x^{4} z^{2}}{4 y^{5}}) {(\frac{2 x^{3} y^{2}}{z^{3}})}^{2}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

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

$\frac{x^{4} z^{2}}{4 y^{5}} \cdot \frac{{(2 x^{3} y^{2})}^{2}}{{(z^{3})}^{2}}$

Step 1.2

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

$\frac{x^{4} z^{2}}{4 y^{5}} \cdot \frac{{(2 x^{3})}^{2} {(y^{2})}^{2}}{{(z^{3})}^{2}}$

Step 1.3

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

$\frac{x^{4} z^{2}}{4 y^{5}} \cdot \frac{2^{2} {(x^{3})}^{2} {(y^{2})}^{2}}{{(z^{3})}^{2}}$

Step 2

Combine.

$\frac{x^{4} z^{2} (2^{2} {(x^{3})}^{2} {(y^{2})}^{2})}{4 y^{5} {(z^{3})}^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{x^{4} z^{2} \cdot 4 {(x^{3})}^{2} {(y^{2})}^{2}}{4 y^{5} {(z^{3})}^{2}}$

Step 3.2

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

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Step 3.2.1

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

$\frac{x^{4} z^{2} \cdot 4 x^{3 \cdot 2} {(y^{2})}^{2}}{4 y^{5} {(z^{3})}^{2}}$

Step 3.2.2

Multiply $3$ by $2$ .

$\frac{x^{4} z^{2} \cdot 4 x^{6} {(y^{2})}^{2}}{4 y^{5} {(z^{3})}^{2}}$

Step 3.3

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

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Step 3.3.1

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

$\frac{x^{4} z^{2} \cdot 4 x^{6} y^{2 \cdot 2}}{4 y^{5} {(z^{3})}^{2}}$

Step 3.3.2

Multiply $2$ by $2$ .

$\frac{x^{4} z^{2} \cdot 4 x^{6} y^{4}}{4 y^{5} {(z^{3})}^{2}}$

Step 3.4

Multiply $x^{4}$ by $x^{6}$ by adding the exponents.

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Step 3.4.1

Move $x^{6}$ .

$\frac{x^{6} x^{4} z^{2} \cdot 4 y^{4}}{4 y^{5} {(z^{3})}^{2}}$

Step 3.4.2

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

$\frac{x^{6 + 4} z^{2} \cdot 4 y^{4}}{4 y^{5} {(z^{3})}^{2}}$

Step 3.4.3

Add $6$ and $4$ .

$\frac{x^{10} z^{2} \cdot 4 y^{4}}{4 y^{5} {(z^{3})}^{2}}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

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

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Step 4.1.1

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

$\frac{x^{10} z^{2} \cdot 4 y^{4}}{4 y^{5} z^{3 \cdot 2}}$

Step 4.1.2

Multiply $3$ by $2$ .

$\frac{x^{10} z^{2} \cdot 4 y^{4}}{4 y^{5} z^{6}}$

Step 4.2

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

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Step 4.2.1

Factor $z^{2}$ out of $x^{10} z^{2} \cdot 4 y^{4}$ .

$\frac{z^{2} (x^{10} \cdot 4 y^{4})}{4 y^{5} z^{6}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $z^{2}$ out of $4 y^{5} z^{6}$ .

$\frac{z^{2} (x^{10} \cdot 4 y^{4})}{z^{2} (4 y^{5} z^{4})}$

Step 4.2.2.2

Cancel the common factor.

$\frac{z^{2} (x^{10} \cdot 4 y^{4})}{z^{2} (4 y^{5} z^{4})}$

Step 4.2.2.3

Rewrite the expression.

$\frac{x^{10} \cdot 4 y^{4}}{4 y^{5} z^{4}}$

Step 4.3

Cancel the common factor of $4$ .

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Step 4.3.1

Cancel the common factor.

$\frac{x^{10} \cdot 4 y^{4}}{4 y^{5} z^{4}}$

Step 4.3.2

Rewrite the expression.

$\frac{x^{10} y^{4}}{y^{5} z^{4}}$

Step 4.4

Cancel the common factor of $y^{4}$ and $y^{5}$ .

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Step 4.4.1

Factor $y^{4}$ out of $x^{10} y^{4}$ .

$\frac{y^{4} x^{10}}{y^{5} z^{4}}$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $y^{4}$ out of $y^{5} z^{4}$ .

$\frac{y^{4} x^{10}}{y^{4} (y z^{4})}$

Step 4.4.2.2

Cancel the common factor.

$\frac{y^{4} x^{10}}{y^{4} (y z^{4})}$

Step 4.4.2.3

Rewrite the expression.

$\frac{x^{10}}{y z^{4}}$