Simplify (2x^-2y^3)^4*2x^19y^-6

${(2 x^{- 2} y^{3})}^{4} \cdot 2 x^{19} y^{- 6}$

Step 1

Rewrite using the commutative property of multiplication.

$2 {(2 x^{- 2} y^{3})}^{4} x^{19} y^{- 6}$

Step 2

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

$2 {(2 \frac{1}{x^{2}} y^{3})}^{4} x^{19} y^{- 6}$

Step 3

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

$2 {(\frac{2}{x^{2}} y^{3})}^{4} x^{19} y^{- 6}$

Step 4

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

$2 {(\frac{2 y^{3}}{x^{2}})}^{4} x^{19} y^{- 6}$

Step 5

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

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

$2 \frac{{(2 y^{3})}^{4}}{{(x^{2})}^{4}} x^{19} y^{- 6}$

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

$2 \frac{2^{4} {(y^{3})}^{4}}{{(x^{2})}^{4}} x^{19} y^{- 6}$

Step 6

Simplify the numerator.

Raise $2$ to the power of $4$ .

$2 \frac{16 {(y^{3})}^{4}}{{(x^{2})}^{4}} x^{19} y^{- 6}$

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

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

$2 \frac{16 y^{3 \cdot 4}}{{(x^{2})}^{4}} x^{19} y^{- 6}$

Multiply $3$ by $4$ .

$2 \frac{16 y^{12}}{{(x^{2})}^{4}} x^{19} y^{- 6}$

Step 7

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

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

$2 \frac{16 y^{12}}{x^{2 \cdot 4}} x^{19} y^{- 6}$

Multiply $2$ by $4$ .

$2 \frac{16 y^{12}}{x^{8}} x^{19} y^{- 6}$

Step 8

Multiply $2 \frac{16 y^{12}}{x^{8}}$ .

Combine $2$ and $\frac{16 y^{12}}{x^{8}}$ .

$\frac{2 (16 y^{12})}{x^{8}} x^{19} y^{- 6}$

Multiply $16$ by $2$ .

$\frac{32 y^{12}}{x^{8}} x^{19} y^{- 6}$

Step 9

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

Factor $x^{8}$ out of $x^{19}$ .

$\frac{32 y^{12}}{x^{8}} (x^{8} x^{11}) y^{- 6}$

Cancel the common factor.

$\frac{32 y^{12}}{x^{8}} (x^{8} x^{11}) y^{- 6}$

Rewrite the expression.

$32 y^{12} x^{11} y^{- 6}$

Step 10

Multiply $y^{12}$ by $y^{- 6}$ by adding the exponents.

Move $y^{- 6}$ .

$32 (y^{- 6} y^{12}) x^{11}$

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

$32 y^{- 6 + 12} x^{11}$

Add $- 6$ and $12$ .

$32 y^{6} x^{11}$

Simplify (2x^-2y^3)^4*2x^19y^-6

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