Simplify (4a^2b)^4((-a^3)/(2b))^2

${(4 a^{2} b)}^{4} {(\frac{- a^{3}}{2 b})}^{2}$

Step 1

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

Apply the product rule to $4 a^{2} b$ .

${(4 a^{2})}^{4} b^{4} {(\frac{- a^{3}}{2 b})}^{2}$

Apply the product rule to $4 a^{2}$ .

$4^{4} {(a^{2})}^{4} b^{4} {(\frac{- a^{3}}{2 b})}^{2}$

Step 2

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

$256 {(a^{2})}^{4} b^{4} {(\frac{- a^{3}}{2 b})}^{2}$

Step 3

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

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

$256 a^{2 \cdot 4} b^{4} {(\frac{- a^{3}}{2 b})}^{2}$

Multiply $2$ by $4$ .

$256 a^{8} b^{4} {(\frac{- a^{3}}{2 b})}^{2}$

Step 4

Move the negative in front of the fraction.

$256 a^{8} b^{4} {(- \frac{a^{3}}{2 b})}^{2}$

Step 5

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

Apply the product rule to $- \frac{a^{3}}{2 b}$ .

$256 a^{8} b^{4} ({(- 1)}^{2} {(\frac{a^{3}}{2 b})}^{2})$

Apply the product rule to $\frac{a^{3}}{2 b}$ .

$256 a^{8} b^{4} ({(- 1)}^{2} \frac{{(a^{3})}^{2}}{{(2 b)}^{2}})$

Apply the product rule to $2 b$ .

$256 a^{8} b^{4} ({(- 1)}^{2} \frac{{(a^{3})}^{2}}{2^{2} b^{2}})$

Step 6

Simplify the expression.

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

$256 a^{8} b^{4} (1 \frac{{(a^{3})}^{2}}{2^{2} b^{2}})$

Multiply $\frac{{(a^{3})}^{2}}{2^{2} b^{2}}$ by $1$ .

$256 a^{8} b^{4} \frac{{(a^{3})}^{2}}{2^{2} b^{2}}$

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

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

$256 a^{8} b^{4} \frac{a^{3 \cdot 2}}{2^{2} b^{2}}$

Multiply $3$ by $2$ .

$256 a^{8} b^{4} \frac{a^{6}}{2^{2} b^{2}}$

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

$256 a^{8} b^{4} \frac{a^{6}}{4 b^{2}}$

Step 7

Cancel the common factor of $4 b^{2}$ .

Factor $4 b^{2}$ out of $256 a^{8} b^{4}$ .

$4 b^{2} (64 a^{8} b^{2}) \frac{a^{6}}{4 b^{2}}$

Cancel the common factor.

$4 b^{2} (64 a^{8} b^{2}) \frac{a^{6}}{4 b^{2}}$

Rewrite the expression.

$64 a^{8} b^{2} a^{6}$

Step 8

Multiply $a^{8}$ by $a^{6}$ by adding the exponents.

Move $a^{6}$ .

$64 (a^{6} a^{8}) b^{2}$

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

$64 a^{6 + 8} b^{2}$

Add $6$ and $8$ .

$64 a^{14} b^{2}$

Simplify (4a^2b)^4((-a^3)/(2b))^2

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