Simplify (12x^2y^-2)^5(4xy^-3)^-7

${(12 x^{2} y^{- 2})}^{5} {(4 x y^{- 3})}^{- 7}$

Step 1

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

${(12 x^{2} \frac{1}{y^{2}})}^{5} {(4 x y^{- 3})}^{- 7}$

Step 2

Multiply $12 x^{2} \frac{1}{y^{2}}$ .

Combine $12$ and $\frac{1}{y^{2}}$ .

${(x^{2} \frac{12}{y^{2}})}^{5} {(4 x y^{- 3})}^{- 7}$

Combine $x^{2}$ and $\frac{12}{y^{2}}$ .

${(\frac{x^{2} \cdot 12}{y^{2}})}^{5} {(4 x y^{- 3})}^{- 7}$

Step 3

Move $12$ to the left of $x^{2}$ .

${(\frac{12 \cdot x^{2}}{y^{2}})}^{5} {(4 x y^{- 3})}^{- 7}$

Step 4

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

Apply the product rule to $\frac{12 x^{2}}{y^{2}}$ .

$\frac{{(12 x^{2})}^{5}}{{(y^{2})}^{5}} {(4 x y^{- 3})}^{- 7}$

Apply the product rule to $12 x^{2}$ .

$\frac{12^{5} {(x^{2})}^{5}}{{(y^{2})}^{5}} {(4 x y^{- 3})}^{- 7}$

Step 5

Simplify the numerator.

Raise $12$ to the power of $5$ .

$\frac{248832 {(x^{2})}^{5}}{{(y^{2})}^{5}} {(4 x y^{- 3})}^{- 7}$

Multiply the exponents in ${(x^{2})}^{5}$ .

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

$\frac{248832 x^{2 \cdot 5}}{{(y^{2})}^{5}} {(4 x y^{- 3})}^{- 7}$

Multiply $2$ by $5$ .

$\frac{248832 x^{10}}{{(y^{2})}^{5}} {(4 x y^{- 3})}^{- 7}$

Step 6

Multiply the exponents in ${(y^{2})}^{5}$ .

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

$\frac{248832 x^{10}}{y^{2 \cdot 5}} {(4 x y^{- 3})}^{- 7}$

Multiply $2$ by $5$ .

$\frac{248832 x^{10}}{y^{10}} {(4 x y^{- 3})}^{- 7}$

Step 7

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

$\frac{248832 x^{10}}{y^{10}} {(4 x \frac{1}{y^{3}})}^{- 7}$

Step 8

Multiply $4 x \frac{1}{y^{3}}$ .

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

$\frac{248832 x^{10}}{y^{10}} {(x \frac{4}{y^{3}})}^{- 7}$

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

$\frac{248832 x^{10}}{y^{10}} {(\frac{x \cdot 4}{y^{3}})}^{- 7}$

Step 9

Move $4$ to the left of $x$ .

$\frac{248832 x^{10}}{y^{10}} {(\frac{4 \cdot x}{y^{3}})}^{- 7}$

Step 10

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

$\frac{248832 x^{10}}{y^{10}} {(\frac{y^{3}}{4 x})}^{7}$

Step 11

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

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

$\frac{248832 x^{10}}{y^{10}} \cdot \frac{{(y^{3})}^{7}}{{(4 x)}^{7}}$

Apply the product rule to $4 x$ .

$\frac{248832 x^{10}}{y^{10}} \cdot \frac{{(y^{3})}^{7}}{4^{7} x^{7}}$

Step 12

Combine.

$\frac{248832 x^{10} {(y^{3})}^{7}}{y^{10} (4^{7} x^{7})}$

Step 13

Cancel the common factor of $x^{10}$ and $x^{7}$ .

Factor $x^{7}$ out of $248832 x^{10} {(y^{3})}^{7}$ .

$\frac{x^{7} (248832 x^{3} {(y^{3})}^{7})}{y^{10} (4^{7} x^{7})}$

Cancel the common factors.

Factor $x^{7}$ out of $y^{10} (4^{7} x^{7})$ .

$\frac{x^{7} (248832 x^{3} {(y^{3})}^{7})}{x^{7} (y^{10} \cdot (4^{7}))}$

Cancel the common factor.

$\frac{x^{7} (248832 x^{3} {(y^{3})}^{7})}{x^{7} (y^{10} \cdot (4^{7}))}$

Rewrite the expression.

$\frac{248832 x^{3} {(y^{3})}^{7}}{y^{10} \cdot (4^{7})}$

Step 14

Simplify the expression.

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

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

$\frac{248832 x^{3} y^{3 \cdot 7}}{y^{10} \cdot 4^{7}}$

Multiply $3$ by $7$ .

$\frac{248832 x^{3} y^{21}}{y^{10} \cdot 4^{7}}$

Raise $4$ to the power of $7$ .

$\frac{248832 x^{3} y^{21}}{y^{10} \cdot 16384}$

Step 15

Cancel the common factor of $248832$ and $16384$ .

Factor $1024$ out of $248832 x^{3} y^{21}$ .

$\frac{1024 (243 x^{3} y^{21})}{y^{10} \cdot 16384}$

Cancel the common factors.

Factor $1024$ out of $y^{10} \cdot 16384$ .

$\frac{1024 (243 x^{3} y^{21})}{1024 (y^{10} \cdot 16)}$

Cancel the common factor.

$\frac{1024 (243 x^{3} y^{21})}{1024 (y^{10} \cdot 16)}$

Rewrite the expression.

$\frac{243 x^{3} y^{21}}{y^{10} \cdot 16}$

Step 16

Cancel the common factor of $y^{21}$ and $y^{10}$ .

Factor $y^{10}$ out of $243 x^{3} y^{21}$ .

$\frac{y^{10} (243 x^{3} y^{11})}{y^{10} \cdot 16}$

Cancel the common factors.

Factor $y^{10}$ out of $y^{10} \cdot 16$ .

$\frac{y^{10} (243 x^{3} y^{11})}{y^{10} (16)}$

Cancel the common factor.

$\frac{y^{10} (243 x^{3} y^{11})}{y^{10} \cdot 16}$

Rewrite the expression.

$\frac{243 x^{3} y^{11}}{16}$

Simplify (12x^2y^-2)^5(4xy^-3)^-7

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