Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $- 6 x^{4} y^{6}$ .

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

Step 1.2

Apply the product rule to $- 6 x^{4}$ .

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

Step 1.3

Raise $- 6$ to the power of $2$ .

$\frac{36 {(x^{4})}^{2} {(y^{6})}^{2}}{{(- 4 x^{- 3} y^{5})}^{3}}$

Step 1.4

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

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Step 1.4.1

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

$\frac{36 x^{4 \cdot 2} {(y^{6})}^{2}}{{(- 4 x^{- 3} y^{5})}^{3}}$

Step 1.4.2

Multiply $4$ by $2$ .

$\frac{36 x^{8} {(y^{6})}^{2}}{{(- 4 x^{- 3} y^{5})}^{3}}$

Step 1.5

Multiply the exponents in ${(y^{6})}^{2}$ .

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Step 1.5.1

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

$\frac{36 x^{8} y^{6 \cdot 2}}{{(- 4 x^{- 3} y^{5})}^{3}}$

Step 1.5.2

Multiply $6$ by $2$ .

$\frac{36 x^{8} y^{12}}{{(- 4 x^{- 3} y^{5})}^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $- 4 x^{- 3} y^{5}$ .

$\frac{36 x^{8} y^{12}}{{(- 4 x^{- 3})}^{3} {(y^{5})}^{3}}$

Step 2.2

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

$\frac{36 x^{8} y^{12}}{{(- 4 \frac{1}{x^{3}})}^{3} {(y^{5})}^{3}}$

Step 2.3

Combine $- 4$ and $\frac{1}{x^{3}}$ .

$\frac{36 x^{8} y^{12}}{{(\frac{- 4}{x^{3}})}^{3} {(y^{5})}^{3}}$

Step 2.4

Move the negative in front of the fraction.

$\frac{36 x^{8} y^{12}}{{(- \frac{4}{x^{3}})}^{3} {(y^{5})}^{3}}$

Step 2.5

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

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Step 2.5.1

Apply the product rule to $- \frac{4}{x^{3}}$ .

$\frac{36 x^{8} y^{12}}{{(- 1)}^{3} {(\frac{4}{x^{3}})}^{3} {(y^{5})}^{3}}$

Step 2.5.2

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

$\frac{36 x^{8} y^{12}}{{(- 1)}^{3} \frac{4^{3}}{{(x^{3})}^{3}} {(y^{5})}^{3}}$

Step 2.6

Raise $- 1$ to the power of $3$ .

$\frac{36 x^{8} y^{12}}{- \frac{4^{3}}{{(x^{3})}^{3}} {(y^{5})}^{3}}$

Step 2.7

Raise $4$ to the power of $3$ .

$\frac{36 x^{8} y^{12}}{- \frac{64}{{(x^{3})}^{3}} {(y^{5})}^{3}}$

Step 2.8

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

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Step 2.8.1

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

$\frac{36 x^{8} y^{12}}{- \frac{64}{x^{3 \cdot 3}} {(y^{5})}^{3}}$

Step 2.8.2

Multiply $3$ by $3$ .

$\frac{36 x^{8} y^{12}}{- \frac{64}{x^{9}} {(y^{5})}^{3}}$

Step 2.9

Multiply the exponents in ${(y^{5})}^{3}$ .

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Step 2.9.1

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

$\frac{36 x^{8} y^{12}}{- \frac{64}{x^{9}} y^{5 \cdot 3}}$

Step 2.9.2

Multiply $5$ by $3$ .

$\frac{36 x^{8} y^{12}}{- \frac{64}{x^{9}} y^{15}}$

Step 2.10

Combine $y^{15}$ and $\frac{64}{x^{9}}$ .

$\frac{36 x^{8} y^{12}}{- \frac{y^{15} \cdot 64}{x^{9}}}$

Step 2.11

Move $64$ to the left of $y^{15}$ .

$\frac{36 x^{8} y^{12}}{- \frac{64 y^{15}}{x^{9}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$36 x^{8} y^{12} (- \frac{x^{9}}{64 y^{15}})$

Step 4

Cancel the common factor of $4 y^{12}$ .

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Step 4.1

Move the leading negative in $- \frac{x^{9}}{64 y^{15}}$ into the numerator.

$36 x^{8} y^{12} \frac{- x^{9}}{64 y^{15}}$

Step 4.2

Factor $4 y^{12}$ out of $36 x^{8} y^{12}$ .

$4 y^{12} (9 x^{8}) \frac{- x^{9}}{64 y^{15}}$

Step 4.3

Factor $4 y^{12}$ out of $64 y^{15}$ .

$4 y^{12} (9 x^{8}) \frac{- x^{9}}{4 y^{12} (16 y^{3})}$

Step 4.4

Cancel the common factor.

$4 y^{12} (9 x^{8}) \frac{- x^{9}}{4 y^{12} (16 y^{3})}$

Step 4.5

Rewrite the expression.

$9 x^{8} \frac{- x^{9}}{16 y^{3}}$

Step 5

Combine $9$ and $\frac{- x^{9}}{16 y^{3}}$ .

$x^{8} \frac{9 (- x^{9})}{16 y^{3}}$

Step 6

Multiply $- 1$ by $9$ .

$x^{8} \frac{- 9 x^{9}}{16 y^{3}}$

Step 7

Combine $x^{8}$ and $\frac{- 9 x^{9}}{16 y^{3}}$ .

$\frac{x^{8} (- 9 x^{9})}{16 y^{3}}$

Step 8

Multiply $x^{8}$ by $x^{9}$ by adding the exponents.

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Step 8.1

Move $x^{9}$ .

$\frac{x^{9} x^{8} \cdot - 9}{16 y^{3}}$

Step 8.2

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

$\frac{x^{9 + 8} \cdot - 9}{16 y^{3}}$

Step 8.3

Add $9$ and $8$ .

$\frac{x^{17} \cdot - 9}{16 y^{3}}$

Step 9

Move $- 9$ to the left of $x^{17}$ .

$\frac{- 9 \cdot x^{17}}{16 y^{3}}$

Step 10

Move the negative in front of the fraction.

$- \frac{9 x^{17}}{16 y^{3}}$