Simplify (a^3b^4c^6)(4ac^3)^2

$(a^{3} b^{4} c^{6}) {(4 a c^{3})}^{2}$

Step 1

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

Apply the product rule to $4 a c^{3}$ .

$a^{3} b^{4} c^{6} ({(4 a)}^{2} {(c^{3})}^{2})$

Apply the product rule to $4 a$ .

$a^{3} b^{4} c^{6} (4^{2} a^{2} {(c^{3})}^{2})$

Step 2

Rewrite using the commutative property of multiplication.

$4^{2} a^{3} b^{4} c^{6} (a^{2} {(c^{3})}^{2})$

Step 3

Multiply $a^{3}$ by $a^{2}$ by adding the exponents.

Move $a^{2}$ .

$4^{2} (a^{2} a^{3}) b^{4} c^{6} {(c^{3})}^{2}$

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

$4^{2} a^{2 + 3} b^{4} c^{6} {(c^{3})}^{2}$

Add $2$ and $3$ .

$4^{2} a^{5} b^{4} c^{6} {(c^{3})}^{2}$

Step 4

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

$16 a^{5} b^{4} c^{6} {(c^{3})}^{2}$

Step 5

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

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

$16 a^{5} b^{4} c^{6} c^{3 \cdot 2}$

Multiply $3$ by $2$ .

$16 a^{5} b^{4} c^{6} c^{6}$

Step 6

Multiply $c^{6}$ by $c^{6}$ by adding the exponents.

Move $c^{6}$ .

$16 a^{5} b^{4} (c^{6} c^{6})$

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

$16 a^{5} b^{4} c^{6 + 6}$

Add $6$ and $6$ .

$16 a^{5} b^{4} c^{12}$