Finite Math Examples

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

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{1}{x^{3}} y^{2})}^{4} {(- 2 x^{3} y^{7})}^{- 3}$

Step 2

Combine $\frac{1}{x^{3}}$ and $y^{2}$ .

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

Multiply the exponents in ${(y^{2})}^{4}$ .

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $4$ .

$\frac{y^{8}}{{(x^{3})}^{4}} {(- 2 x^{3} y^{7})}^{- 3}$

Step 3.3

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

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Step 3.3.1

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

$\frac{y^{8}}{x^{3 \cdot 4}} {(- 2 x^{3} y^{7})}^{- 3}$

Step 3.3.2

Multiply $3$ by $4$ .

$\frac{y^{8}}{x^{12}} {(- 2 x^{3} y^{7})}^{- 3}$

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{y^{8}}{x^{12}} \cdot \frac{1}{{(- 2 x^{3} y^{7})}^{3}}$

Step 5

Combine fractions.

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Step 5.1

Combine.

$\frac{y^{8} \cdot 1}{x^{12} {(- 2 x^{3} y^{7})}^{3}}$

Step 5.2

Multiply $y^{8}$ by $1$ .

$\frac{y^{8}}{x^{12} {(- 2 x^{3} y^{7})}^{3}}$

Step 6

Simplify the denominator.

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Step 6.1

Apply the product rule to $- 2 x^{3} y^{7}$ .

$\frac{y^{8}}{x^{12} {(- 2 x^{3})}^{3} {(y^{7})}^{3}}$

Step 6.2

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

$\frac{y^{8}}{x^{12} ({(- 2)}^{3} {(x^{3})}^{3}) {(y^{7})}^{3}}$

Step 6.3

Raise $- 2$ to the power of $3$ .

$\frac{y^{8}}{x^{12} (- 8 {(x^{3})}^{3}) {(y^{7})}^{3}}$

Step 6.4

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

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Step 6.4.1

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

$\frac{y^{8}}{x^{12} (- 8 x^{3 \cdot 3}) {(y^{7})}^{3}}$

Step 6.4.2

Multiply $3$ by $3$ .

$\frac{y^{8}}{x^{12} (- 8 x^{9}) {(y^{7})}^{3}}$

Step 6.5

Multiply the exponents in ${(y^{7})}^{3}$ .

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Step 6.5.1

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

$\frac{y^{8}}{x^{12} (- 8 x^{9}) y^{7 \cdot 3}}$

Step 6.5.2

Multiply $7$ by $3$ .

$\frac{y^{8}}{x^{12} (- 8 x^{9}) y^{21}}$

$\frac{y^{8}}{x^{12} \cdot - 8 x^{9} y^{21}}$

Step 6.6

Multiply $x^{12}$ by $x^{9}$ by adding the exponents.

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Step 6.6.1

Move $x^{9}$ .

$\frac{y^{8}}{x^{9} x^{12} \cdot - 8 y^{21}}$

Step 6.6.2

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

$\frac{y^{8}}{x^{9 + 12} \cdot - 8 y^{21}}$

Step 6.6.3

Add $9$ and $12$ .

$\frac{y^{8}}{x^{21} \cdot - 8 y^{21}}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $y^{8}$ and $y^{21}$ .

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Step 7.1.1

Multiply by $1$ .

$\frac{y^{8} \cdot 1}{x^{21} \cdot - 8 y^{21}}$

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $y^{8}$ out of $x^{21} \cdot - 8 y^{21}$ .

$\frac{y^{8} \cdot 1}{y^{8} (x^{21} \cdot - 8 y^{13})}$

Step 7.1.2.2

Cancel the common factor.

$\frac{y^{8} \cdot 1}{y^{8} (x^{21} \cdot - 8 y^{13})}$

Step 7.1.2.3

Rewrite the expression.

$\frac{1}{x^{21} \cdot - 8 y^{13}}$

Step 7.2

Simplify the expression.

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Step 7.2.1

Move $- 8$ to the left of $x^{21}$ .

$\frac{1}{- 8 \cdot x^{21} y^{13}}$

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{1}{8 x^{21} y^{13}}$