Simplify (3ab^3z^2)^2(4a^2b)(2a^3c^2)^4

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

Step 1

Rewrite using the commutative property of multiplication.

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

Step 2

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

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

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

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

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

Apply the product rule to $3 a$ .

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

Step 3

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

Move $a^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Step 4

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

$4 (9 a^{4} {(b^{3})}^{2} {(z^{2})}^{2}) b {(2 a^{3} c^{2})}^{4}$

Step 5

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

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

$4 (9 a^{4} b^{3 \cdot 2} {(z^{2})}^{2}) b {(2 a^{3} c^{2})}^{4}$

Multiply $3$ by $2$ .

$4 (9 a^{4} b^{6} {(z^{2})}^{2}) b {(2 a^{3} c^{2})}^{4}$

Step 6

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

Move $b$ .

$4 (9 a^{4} (b \cdot b^{6}) {(z^{2})}^{2}) {(2 a^{3} c^{2})}^{4}$

Multiply $b$ by $b^{6}$ .

Raise $b$ to the power of $1$ .

$4 (9 a^{4} (b^{1} b^{6}) {(z^{2})}^{2}) {(2 a^{3} c^{2})}^{4}$

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

$4 (9 a^{4} b^{1 + 6} {(z^{2})}^{2}) {(2 a^{3} c^{2})}^{4}$

Add $1$ and $6$ .

$4 (9 a^{4} b^{7} {(z^{2})}^{2}) {(2 a^{3} c^{2})}^{4}$

Step 7

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

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

$4 (9 a^{4} b^{7} z^{2 \cdot 2}) {(2 a^{3} c^{2})}^{4}$

Multiply $2$ by $2$ .

$4 (9 a^{4} b^{7} z^{4}) {(2 a^{3} c^{2})}^{4}$

Step 8

Multiply $9$ by $4$ .

$36 (a^{4} b^{7} z^{4}) {(2 a^{3} c^{2})}^{4}$

Step 9

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

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

$36 a^{4} b^{7} z^{4} ({(2 a^{3})}^{4} {(c^{2})}^{4})$

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

$36 a^{4} b^{7} z^{4} (2^{4} {(a^{3})}^{4} {(c^{2})}^{4})$

Step 10

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

$36 a^{4} b^{7} z^{4} (16 {(a^{3})}^{4} {(c^{2})}^{4})$

Step 11

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

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

$36 a^{4} b^{7} z^{4} (16 a^{3 \cdot 4} {(c^{2})}^{4})$

Multiply $3$ by $4$ .

$36 a^{4} b^{7} z^{4} (16 a^{12} {(c^{2})}^{4})$

Step 12

Multiply $a^{4}$ by $a^{12}$ by adding the exponents.

Move $a^{12}$ .

$36 (a^{12} a^{4}) b^{7} z^{4} (16 {(c^{2})}^{4})$

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

$36 a^{12 + 4} b^{7} z^{4} (16 {(c^{2})}^{4})$

Add $12$ and $4$ .

$36 a^{16} b^{7} z^{4} (16 {(c^{2})}^{4})$

Step 13

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

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

$36 a^{16} b^{7} z^{4} (16 c^{2 \cdot 4})$

Multiply $2$ by $4$ .

$36 a^{16} b^{7} z^{4} (16 c^{8})$

Step 14

Multiply $16$ by $36$ .

$576 a^{16} b^{7} z^{4} c^{8}$

Simplify (3ab^3z^2)^2(4a^2b)(2a^3c^2)^4

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