Simplify (2x^-4y^7)^2*(-3x^-3y^-2)^-3

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

Step 1

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

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

Step 2

Combine $2$ and $\frac{1}{x^{4}}$ .

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

Step 3

Combine $\frac{2}{x^{4}}$ and $y^{7}$ .

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

Step 4

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

Apply the product rule to $\frac{2 y^{7}}{x^{4}}$ .

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

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

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

Step 5

Simplify the numerator.

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

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

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

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

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

Multiply $7$ by $2$ .

$\frac{4 y^{14}}{{(x^{4})}^{2}} \cdot {(- 3 x^{- 3} y^{- 2})}^{- 3}$

Step 6

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

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

$\frac{4 y^{14}}{x^{4 \cdot 2}} \cdot {(- 3 x^{- 3} y^{- 2})}^{- 3}$

Multiply $4$ by $2$ .

$\frac{4 y^{14}}{x^{8}} \cdot {(- 3 x^{- 3} y^{- 2})}^{- 3}$

Step 7

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

$\frac{4 y^{14}}{x^{8}} \cdot {(- 3 \frac{1}{x^{3}} y^{- 2})}^{- 3}$

Step 8

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

$\frac{4 y^{14}}{x^{8}} \cdot {(\frac{- 3}{x^{3}} y^{- 2})}^{- 3}$

Step 9

Move the negative in front of the fraction.

$\frac{4 y^{14}}{x^{8}} \cdot {(- \frac{3}{x^{3}} y^{- 2})}^{- 3}$

Step 10

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

$\frac{4 y^{14}}{x^{8}} \cdot {(- \frac{3}{x^{3}} \cdot \frac{1}{y^{2}})}^{- 3}$

Step 11

Multiply $\frac{1}{y^{2}}$ by $\frac{3}{x^{3}}$ .

$\frac{4 y^{14}}{x^{8}} \cdot {(- \frac{3}{y^{2} x^{3}})}^{- 3}$

Step 12

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

$\frac{4 y^{14}}{x^{8}} \cdot {(- \frac{y^{2} x^{3}}{3})}^{3}$

Step 13

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

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

$\frac{4 y^{14}}{x^{8}} \cdot ({(- 1)}^{3} {(\frac{y^{2} x^{3}}{3})}^{3})$

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

$\frac{4 y^{14}}{x^{8}} \cdot ({(- 1)}^{3} \frac{{(y^{2} x^{3})}^{3}}{3^{3}})$

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

$\frac{4 y^{14}}{x^{8}} \cdot ({(- 1)}^{3} \frac{{(y^{2})}^{3} {(x^{3})}^{3}}{3^{3}})$

Step 14

Simplify the expression.

Rewrite using the commutative property of multiplication.

${(- 1)}^{3} \frac{4 y^{14}}{x^{8}} \frac{{(y^{2})}^{3} {(x^{3})}^{3}}{3^{3}}$

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

$- \frac{4 y^{14}}{x^{8}} \cdot \frac{{(y^{2})}^{3} {(x^{3})}^{3}}{3^{3}}$

Step 15

Simplify the numerator.

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

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

$- \frac{4 y^{14}}{x^{8}} \cdot \frac{y^{2 \cdot 3} {(x^{3})}^{3}}{3^{3}}$

Multiply $2$ by $3$ .

$- \frac{4 y^{14}}{x^{8}} \cdot \frac{y^{6} {(x^{3})}^{3}}{3^{3}}$

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

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

$- \frac{4 y^{14}}{x^{8}} \cdot \frac{y^{6} x^{3 \cdot 3}}{3^{3}}$

Multiply $3$ by $3$ .

$- \frac{4 y^{14}}{x^{8}} \cdot \frac{y^{6} x^{9}}{3^{3}}$

Step 16

Simplify terms.

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

$- \frac{4 y^{14}}{x^{8}} \cdot \frac{y^{6} x^{9}}{27}$

Cancel the common factor of $x^{8}$ .

Move the leading negative in $- \frac{4 y^{14}}{x^{8}}$ into the numerator.

$\frac{- 4 y^{14}}{x^{8}} \cdot \frac{y^{6} x^{9}}{27}$

Factor $x^{8}$ out of $y^{6} x^{9}$ .

$\frac{- 4 y^{14}}{x^{8}} \cdot \frac{x^{8} (y^{6} x)}{27}$

Cancel the common factor.

$\frac{- 4 y^{14}}{x^{8}} \cdot \frac{x^{8} (y^{6} x)}{27}$

Rewrite the expression.

$- 4 y^{14} \frac{y^{6} x}{27}$

Combine $- 4$ and $\frac{y^{6} x}{27}$ .

$y^{14} \frac{- 4 (y^{6} x)}{27}$

Combine $y^{14}$ and $\frac{- 4 (y^{6} x)}{27}$ .

$\frac{y^{14} (- 4 (y^{6} x))}{27}$

Step 17

Multiply $y^{14}$ by $y^{6}$ by adding the exponents.

Move $y^{6}$ .

$\frac{y^{6} y^{14} (- 4 (x))}{27}$

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

$\frac{y^{6 + 14} (- 4 (x))}{27}$

Add $6$ and $14$ .

$\frac{y^{20} (- 4 (x))}{27}$

Step 18

Simplify the numerator.

Rewrite.

$\frac{y^{14} (- 4 (y^{6} x))}{27}$

Multiply $y^{14}$ by $y^{6}$ by adding the exponents.

Move $y^{6}$ .

$\frac{y^{6} y^{14} (- 4 (x))}{27}$

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

$\frac{y^{6 + 14} (- 4 (x))}{27}$

Add $6$ and $14$ .

$\frac{y^{20} (- 4 (x))}{27}$

Remove unnecessary parentheses.

$\frac{y^{20} \cdot - 4 x}{27}$

Step 19

Simplify the expression.

Move $- 4$ to the left of $y^{20}$ .

$\frac{- 4 \cdot y^{20} x}{27}$

Move the negative in front of the fraction.

$- \frac{4 y^{20} x}{27}$

Simplify (2x^-4y^7)^2*(-3x^-3y^-2)^-3

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