Simplify (3ab^2c)^2(-2a^2b^4)^2(a^4c^2)^3(a^2b^4c^5)^2(2a^3b^2c^4)^3

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

Step 1

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

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

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

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

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

Apply the product rule to $3 a$ .

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

Step 2

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

$9 a^{2} {(b^{2})}^{2} c^{2} {(- 2 a^{2} b^{4})}^{2} {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 3

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

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

$9 a^{2} b^{2 \cdot 2} c^{2} {(- 2 a^{2} b^{4})}^{2} {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Multiply $2$ by $2$ .

$9 a^{2} b^{4} c^{2} {(- 2 a^{2} b^{4})}^{2} {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 4

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

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

$9 a^{2} b^{4} c^{2} ({(- 2 a^{2})}^{2} {(b^{4})}^{2}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

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

$9 a^{2} b^{4} c^{2} ({(- 2)}^{2} {(a^{2})}^{2} {(b^{4})}^{2}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 5

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

$9 a^{2} b^{4} c^{2} (4 {(a^{2})}^{2} {(b^{4})}^{2}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 6

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

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

$9 a^{2} b^{4} c^{2} (4 a^{2 \cdot 2} {(b^{4})}^{2}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Multiply $2$ by $2$ .

$9 a^{2} b^{4} c^{2} (4 a^{4} {(b^{4})}^{2}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 7

Multiply $a^{2}$ by $a^{4}$ by adding the exponents.

Move $a^{4}$ .

$9 (a^{4} a^{2}) b^{4} c^{2} (4 {(b^{4})}^{2}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

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

$9 a^{4 + 2} b^{4} c^{2} (4 {(b^{4})}^{2}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Add $4$ and $2$ .

$9 a^{6} b^{4} c^{2} (4 {(b^{4})}^{2}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 8

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

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

$9 a^{6} b^{4} c^{2} (4 b^{4 \cdot 2}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Multiply $4$ by $2$ .

$9 a^{6} b^{4} c^{2} (4 b^{8}) {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 9

Multiply $b^{4}$ by $b^{8}$ by adding the exponents.

Move $b^{8}$ .

$9 a^{6} (b^{8} b^{4}) c^{2} \cdot 4 {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

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

$9 a^{6} b^{8 + 4} c^{2} \cdot 4 {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Add $8$ and $4$ .

$9 a^{6} b^{12} c^{2} \cdot 4 {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 10

Multiply $4$ by $9$ .

$36 a^{6} b^{12} c^{2} {(a^{4} c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 11

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

$36 a^{6} b^{12} c^{2} ({(a^{4})}^{3} {(c^{2})}^{3}) {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 12

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

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

$36 a^{6} b^{12} c^{2} (a^{4 \cdot 3} {(c^{2})}^{3}) {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Multiply $4$ by $3$ .

$36 a^{6} b^{12} c^{2} (a^{12} {(c^{2})}^{3}) {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 13

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

Move $a^{12}$ .

$36 (a^{12} a^{6}) b^{12} c^{2} {(c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

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

$36 a^{12 + 6} b^{12} c^{2} {(c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Add $12$ and $6$ .

$36 a^{18} b^{12} c^{2} {(c^{2})}^{3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 14

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

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

$36 a^{18} b^{12} c^{2} c^{2 \cdot 3} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Multiply $2$ by $3$ .

$36 a^{18} b^{12} c^{2} c^{6} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 15

Multiply $c^{2}$ by $c^{6}$ by adding the exponents.

Move $c^{6}$ .

$36 a^{18} b^{12} (c^{6} c^{2}) {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

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

$36 a^{18} b^{12} c^{6 + 2} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Add $6$ and $2$ .

$36 a^{18} b^{12} c^{8} {(a^{2} b^{4} c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 16

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

Apply the product rule to $a^{2} b^{4} c^{5}$ .

$36 a^{18} b^{12} c^{8} ({(a^{2} b^{4})}^{2} {(c^{5})}^{2}) {(2 a^{3} b^{2} c^{4})}^{3}$

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

$36 a^{18} b^{12} c^{8} ({(a^{2})}^{2} {(b^{4})}^{2} {(c^{5})}^{2}) {(2 a^{3} b^{2} c^{4})}^{3}$

Step 17

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

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

$36 a^{18} b^{12} c^{8} (a^{2 \cdot 2} {(b^{4})}^{2} {(c^{5})}^{2}) {(2 a^{3} b^{2} c^{4})}^{3}$

Multiply $2$ by $2$ .

$36 a^{18} b^{12} c^{8} (a^{4} {(b^{4})}^{2} {(c^{5})}^{2}) {(2 a^{3} b^{2} c^{4})}^{3}$

Step 18

Multiply $a^{18}$ by $a^{4}$ by adding the exponents.

Move $a^{4}$ .

$36 (a^{4} a^{18}) b^{12} c^{8} ({(b^{4})}^{2} {(c^{5})}^{2}) {(2 a^{3} b^{2} c^{4})}^{3}$

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

$36 a^{4 + 18} b^{12} c^{8} ({(b^{4})}^{2} {(c^{5})}^{2}) {(2 a^{3} b^{2} c^{4})}^{3}$

Add $4$ and $18$ .

$36 a^{22} b^{12} c^{8} ({(b^{4})}^{2} {(c^{5})}^{2}) {(2 a^{3} b^{2} c^{4})}^{3}$

Step 19

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

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

$36 a^{22} b^{12} c^{8} (b^{4 \cdot 2} {(c^{5})}^{2}) {(2 a^{3} b^{2} c^{4})}^{3}$

Multiply $4$ by $2$ .

$36 a^{22} b^{12} c^{8} (b^{8} {(c^{5})}^{2}) {(2 a^{3} b^{2} c^{4})}^{3}$

Step 20

Multiply $b^{12}$ by $b^{8}$ by adding the exponents.

Move $b^{8}$ .

$36 a^{22} (b^{8} b^{12}) c^{8} {(c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

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

$36 a^{22} b^{8 + 12} c^{8} {(c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Add $8$ and $12$ .

$36 a^{22} b^{20} c^{8} {(c^{5})}^{2} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 21

Multiply the exponents in ${(c^{5})}^{2}$ .

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

$36 a^{22} b^{20} c^{8} c^{5 \cdot 2} {(2 a^{3} b^{2} c^{4})}^{3}$

Multiply $5$ by $2$ .

$36 a^{22} b^{20} c^{8} c^{10} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 22

Multiply $c^{8}$ by $c^{10}$ by adding the exponents.

Move $c^{10}$ .

$36 a^{22} b^{20} (c^{10} c^{8}) {(2 a^{3} b^{2} c^{4})}^{3}$

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

$36 a^{22} b^{20} c^{10 + 8} {(2 a^{3} b^{2} c^{4})}^{3}$

Add $10$ and $8$ .

$36 a^{22} b^{20} c^{18} {(2 a^{3} b^{2} c^{4})}^{3}$

Step 23

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

Apply the product rule to $2 a^{3} b^{2} c^{4}$ .

$36 a^{22} b^{20} c^{18} ({(2 a^{3} b^{2})}^{3} {(c^{4})}^{3})$

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

$36 a^{22} b^{20} c^{18} ({(2 a^{3})}^{3} {(b^{2})}^{3} {(c^{4})}^{3})$

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

$36 a^{22} b^{20} c^{18} (2^{3} {(a^{3})}^{3} {(b^{2})}^{3} {(c^{4})}^{3})$

Step 24

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

$36 a^{22} b^{20} c^{18} (8 {(a^{3})}^{3} {(b^{2})}^{3} {(c^{4})}^{3})$

Step 25

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

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

$36 a^{22} b^{20} c^{18} (8 a^{3 \cdot 3} {(b^{2})}^{3} {(c^{4})}^{3})$

Multiply $3$ by $3$ .

$36 a^{22} b^{20} c^{18} (8 a^{9} {(b^{2})}^{3} {(c^{4})}^{3})$

Step 26

Multiply $a^{22}$ by $a^{9}$ by adding the exponents.

Move $a^{9}$ .

$36 (a^{9} a^{22}) b^{20} c^{18} (8 {(b^{2})}^{3} {(c^{4})}^{3})$

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

$36 a^{9 + 22} b^{20} c^{18} (8 {(b^{2})}^{3} {(c^{4})}^{3})$

Add $9$ and $22$ .

$36 a^{31} b^{20} c^{18} (8 {(b^{2})}^{3} {(c^{4})}^{3})$

Step 27

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

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

$36 a^{31} b^{20} c^{18} (8 b^{2 \cdot 3} {(c^{4})}^{3})$

Multiply $2$ by $3$ .

$36 a^{31} b^{20} c^{18} (8 b^{6} {(c^{4})}^{3})$

Step 28

Multiply $b^{20}$ by $b^{6}$ by adding the exponents.

Move $b^{6}$ .

$36 a^{31} (b^{6} b^{20}) c^{18} (8 {(c^{4})}^{3})$

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

$36 a^{31} b^{6 + 20} c^{18} (8 {(c^{4})}^{3})$

Add $6$ and $20$ .

$36 a^{31} b^{26} c^{18} (8 {(c^{4})}^{3})$

Step 29

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

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

$36 a^{31} b^{26} c^{18} (8 c^{4 \cdot 3})$

Multiply $4$ by $3$ .

$36 a^{31} b^{26} c^{18} (8 c^{12})$

Step 30

Multiply $c^{18}$ by $c^{12}$ by adding the exponents.

Move $c^{12}$ .

$36 a^{31} b^{26} (c^{12} c^{18}) \cdot 8$

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

$36 a^{31} b^{26} c^{12 + 18} \cdot 8$

Add $12$ and $18$ .

$36 a^{31} b^{26} c^{30} \cdot 8$

Step 31

Multiply $8$ by $36$ .

$288 a^{31} b^{26} c^{30}$

Simplify (3ab^2c)^2(-2a^2b^4)^2(a^4c^2)^3(a^2b^4c^5)^2(2a^3b^2c^4)^3

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