Simplify (9a^3b^4c)^2(2a^2b^3)^3(ac)

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

Step 1

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

Apply the product rule to $9 a^{3} b^{4} c$ .

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

Apply the product rule to $9 a^{3} b^{4}$ .

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

Apply the product rule to $9 a^{3}$ .

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

Step 2

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

Move $c$ .

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

Multiply $c$ by $c^{2}$ .

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

$9^{2} {(a^{3})}^{2} {(b^{4})}^{2} (c^{1} c^{2}) \cdot {(2 a^{2} b^{3})}^{3} \cdot a$

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

$9^{2} {(a^{3})}^{2} {(b^{4})}^{2} c^{1 + 2} \cdot {(2 a^{2} b^{3})}^{3} \cdot a$

Add $1$ and $2$ .

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

Step 3

Raise $9$ to the power of $2$ .

$81 {(a^{3})}^{2} {(b^{4})}^{2} c^{3} \cdot {(2 a^{2} b^{3})}^{3} \cdot a$

Step 4

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

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

$81 a^{3 \cdot 2} {(b^{4})}^{2} c^{3} \cdot {(2 a^{2} b^{3})}^{3} \cdot a$

Multiply $3$ by $2$ .

$81 a^{6} {(b^{4})}^{2} c^{3} \cdot {(2 a^{2} b^{3})}^{3} \cdot a$

Step 5

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

Move $a$ .

$81 (a \cdot a^{6}) {(b^{4})}^{2} c^{3} \cdot {(2 a^{2} b^{3})}^{3}$

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

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

$81 (a^{1} a^{6}) {(b^{4})}^{2} c^{3} \cdot {(2 a^{2} b^{3})}^{3}$

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

$81 a^{1 + 6} {(b^{4})}^{2} c^{3} \cdot {(2 a^{2} b^{3})}^{3}$

Add $1$ and $6$ .

$81 a^{7} {(b^{4})}^{2} c^{3} \cdot {(2 a^{2} b^{3})}^{3}$

Step 6

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

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

$81 a^{7} b^{4 \cdot 2} c^{3} \cdot {(2 a^{2} b^{3})}^{3}$

Multiply $4$ by $2$ .

$81 a^{7} b^{8} c^{3} \cdot {(2 a^{2} b^{3})}^{3}$

Step 7

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

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

$81 a^{7} b^{8} c^{3} \cdot ({(2 a^{2})}^{3} {(b^{3})}^{3})$

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

$81 a^{7} b^{8} c^{3} \cdot (2^{3} {(a^{2})}^{3} {(b^{3})}^{3})$

Step 8

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

$81 a^{7} b^{8} c^{3} \cdot (8 {(a^{2})}^{3} {(b^{3})}^{3})$

Step 9

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

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

$81 a^{7} b^{8} c^{3} \cdot (8 a^{2 \cdot 3} {(b^{3})}^{3})$

Multiply $2$ by $3$ .

$81 a^{7} b^{8} c^{3} \cdot (8 a^{6} {(b^{3})}^{3})$

Step 10

Multiply $a^{7}$ by $a^{6}$ by adding the exponents.

Move $a^{6}$ .

$81 (a^{6} a^{7}) b^{8} c^{3} \cdot (8 {(b^{3})}^{3})$

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

$81 a^{6 + 7} b^{8} c^{3} \cdot (8 {(b^{3})}^{3})$

Add $6$ and $7$ .

$81 a^{13} b^{8} c^{3} \cdot (8 {(b^{3})}^{3})$

Step 11

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

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

$81 a^{13} b^{8} c^{3} \cdot (8 b^{3 \cdot 3})$

Multiply $3$ by $3$ .

$81 a^{13} b^{8} c^{3} \cdot (8 b^{9})$

Step 12

Multiply $b^{8}$ by $b^{9}$ by adding the exponents.

Move $b^{9}$ .

$81 a^{13} (b^{9} b^{8}) c^{3} \cdot 8$

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

$81 a^{13} b^{9 + 8} c^{3} \cdot 8$

Add $9$ and $8$ .

$81 a^{13} b^{17} c^{3} \cdot 8$

Step 13

Multiply $8$ by $81$ .

$648 a^{13} b^{17} c^{3}$

Simplify (9a^3b^4c)^2*(2a^2b^3)^3*(ac)

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Simplify (9a^3b^4c)^2(2a^2b^3)^3(ac)