Pre-Algebra Examples

${(\frac{20 x^{2} y^{- 3} z^{3}}{5 x^{5} y^{4} z^{6}})}^{- 2}$

Step 1

Move $y^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{20 x^{2} z^{3}}{5 x^{5} y^{4} z^{6} y^{3}})}^{- 2}$

Step 2

Multiply $y^{4}$ by $y^{3}$ by adding the exponents.

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Step 2.1

Move $y^{3}$ .

${(\frac{20 x^{2} z^{3}}{5 x^{5} (y^{3} y^{4}) z^{6}})}^{- 2}$

Step 2.2

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

${(\frac{20 x^{2} z^{3}}{5 x^{5} y^{3 + 4} z^{6}})}^{- 2}$

Step 2.3

Add $3$ and $4$ .

${(\frac{20 x^{2} z^{3}}{5 x^{5} y^{7} z^{6}})}^{- 2}$

Step 3

Cancel the common factor of $20$ and $5$ .

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Step 3.1

Factor $5$ out of $20 x^{2} z^{3}$ .

${(\frac{5 (4 x^{2} z^{3})}{5 x^{5} y^{7} z^{6}})}^{- 2}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $5$ out of $5 x^{5} y^{7} z^{6}$ .

${(\frac{5 (4 x^{2} z^{3})}{5 (x^{5} y^{7} z^{6})})}^{- 2}$

Step 3.2.2

Cancel the common factor.

${(\frac{5 (4 x^{2} z^{3})}{5 (x^{5} y^{7} z^{6})})}^{- 2}$

Step 3.2.3

Rewrite the expression.

${(\frac{4 x^{2} z^{3}}{x^{5} y^{7} z^{6}})}^{- 2}$

Step 4

Cancel the common factor of $x^{2}$ and $x^{5}$ .

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Step 4.1

Factor $x^{2}$ out of $4 x^{2} z^{3}$ .

${(\frac{x^{2} (4 z^{3})}{x^{5} y^{7} z^{6}})}^{- 2}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $x^{2}$ out of $x^{5} y^{7} z^{6}$ .

${(\frac{x^{2} (4 z^{3})}{x^{2} (x^{3} y^{7} z^{6})})}^{- 2}$

Step 4.2.2

Cancel the common factor.

${(\frac{x^{2} (4 z^{3})}{x^{2} (x^{3} y^{7} z^{6})})}^{- 2}$

Step 4.2.3

Rewrite the expression.

${(\frac{4 z^{3}}{x^{3} y^{7} z^{6}})}^{- 2}$

Step 5

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

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Step 5.1

Factor $z^{3}$ out of $4 z^{3}$ .

${(\frac{z^{3} \cdot 4}{x^{3} y^{7} z^{6}})}^{- 2}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $z^{3}$ out of $x^{3} y^{7} z^{6}$ .

${(\frac{z^{3} \cdot 4}{z^{3} (x^{3} y^{7} z^{3})})}^{- 2}$

Step 5.2.2

Cancel the common factor.

${(\frac{z^{3} \cdot 4}{z^{3} (x^{3} y^{7} z^{3})})}^{- 2}$

Step 5.2.3

Rewrite the expression.

${(\frac{4}{x^{3} y^{7} z^{3}})}^{- 2}$

Step 6

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{x^{3} y^{7} z^{3}}{4})}^{2}$

Step 7

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

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Step 7.1

Apply the product rule to $\frac{x^{3} y^{7} z^{3}}{4}$ .

$\frac{{(x^{3} y^{7} z^{3})}^{2}}{4^{2}}$

Step 7.2

Apply the product rule to $x^{3} y^{7} z^{3}$ .

$\frac{{(x^{3} y^{7})}^{2} {(z^{3})}^{2}}{4^{2}}$

Step 7.3

Apply the product rule to $x^{3} y^{7}$ .

$\frac{{(x^{3})}^{2} {(y^{7})}^{2} {(z^{3})}^{2}}{4^{2}}$

Step 8

Simplify the numerator.

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Step 8.1

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

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Step 8.1.1

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

$\frac{x^{3 \cdot 2} {(y^{7})}^{2} {(z^{3})}^{2}}{4^{2}}$

Step 8.1.2

Multiply $3$ by $2$ .

$\frac{x^{6} {(y^{7})}^{2} {(z^{3})}^{2}}{4^{2}}$

Step 8.2

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

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Step 8.2.1

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

$\frac{x^{6} y^{7 \cdot 2} {(z^{3})}^{2}}{4^{2}}$

Step 8.2.2

Multiply $7$ by $2$ .

$\frac{x^{6} y^{14} {(z^{3})}^{2}}{4^{2}}$

Step 8.3

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

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Step 8.3.1

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

$\frac{x^{6} y^{14} z^{3 \cdot 2}}{4^{2}}$

Step 8.3.2

Multiply $3$ by $2$ .

$\frac{x^{6} y^{14} z^{6}}{4^{2}}$

Step 9

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

$\frac{x^{6} y^{14} z^{6}}{16}$