Simplify (2y)^3(-2x^-1)^2(3y^4)

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

Step 1

Rewrite using the commutative property of multiplication.

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

Step 2

Apply the product rule to $2 y$ .

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

Step 3

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

Move $y^{4}$ .

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

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

$3 (2^{3} y^{4 + 3}) {(- 2 x^{- 1})}^{2}$

Add $4$ and $3$ .

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

Step 4

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

$3 (8 y^{7}) {(- 2 x^{- 1})}^{2}$

Step 5

Multiply $8$ by $3$ .

$24 y^{7} {(- 2 x^{- 1})}^{2}$

Step 6

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

$24 y^{7} {(- 2 \frac{1}{x})}^{2}$

Step 7

Combine $- 2$ and $\frac{1}{x}$ .

$24 y^{7} {(\frac{- 2}{x})}^{2}$

Step 8

Move the negative in front of the fraction.

$24 y^{7} {(- \frac{2}{x})}^{2}$

Step 9

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

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

$24 y^{7} ({(- 1)}^{2} {(\frac{2}{x})}^{2})$

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

$24 y^{7} ({(- 1)}^{2} \frac{2^{2}}{x^{2}})$

Step 10

Raise $- 1$ to the power of $2$ .

$24 y^{7} (1 \frac{2^{2}}{x^{2}})$

Step 11

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

$24 y^{7} \frac{2^{2}}{x^{2}}$

Step 12

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

$24 y^{7} \frac{4}{x^{2}}$

Step 13

Multiply $24 y^{7} \frac{4}{x^{2}}$ .

Combine $24$ and $\frac{4}{x^{2}}$ .

$y^{7} \frac{24 \cdot 4}{x^{2}}$

Multiply $24$ by $4$ .

$y^{7} \frac{96}{x^{2}}$

Combine $y^{7}$ and $\frac{96}{x^{2}}$ .

$\frac{y^{7} \cdot 96}{x^{2}}$

Step 14

Move $96$ to the left of $y^{7}$ .

$\frac{96 y^{7}}{x^{2}}$

Simplify (2y)^3(-2x^-1)^2(3y^4)

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