Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

$8 {(a^{2})}^{3} {(b^{3})}^{3} {(c^{4})}^{3} {(- 3 a b)}^{2} {(- b^{3} c)}^{3}$

Step 3

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

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Step 3.1

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

$8 a^{2 \cdot 3} {(b^{3})}^{3} {(c^{4})}^{3} {(- 3 a b)}^{2} {(- b^{3} c)}^{3}$

Step 3.2

Multiply $2$ by $3$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $3$ by $3$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Multiply $4$ by $3$ .

$8 a^{6} b^{9} c^{12} {(- 3 a b)}^{2} {(- b^{3} c)}^{3}$

Step 6

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

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Step 6.1

Apply the product rule to $- 3 a b$ .

$8 a^{6} b^{9} c^{12} ({(- 3 a)}^{2} b^{2}) {(- b^{3} c)}^{3}$

Step 6.2

Apply the product rule to $- 3 a$ .

$8 a^{6} b^{9} c^{12} ({(- 3)}^{2} a^{2} b^{2}) {(- b^{3} c)}^{3}$

Step 7

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

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Step 7.1

Move $a^{2}$ .

$8 (a^{2} a^{6}) b^{9} c^{12} ({(- 3)}^{2} b^{2}) {(- b^{3} c)}^{3}$

Step 7.2

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

$8 a^{2 + 6} b^{9} c^{12} ({(- 3)}^{2} b^{2}) {(- b^{3} c)}^{3}$

Step 7.3

Add $2$ and $6$ .

$8 a^{8} b^{9} c^{12} ({(- 3)}^{2} b^{2}) {(- b^{3} c)}^{3}$

Step 8

Multiply $b^{9}$ by $b^{2}$ by adding the exponents.

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Step 8.1

Move $b^{2}$ .

$8 a^{8} (b^{2} b^{9}) c^{12} {(- 3)}^{2} {(- b^{3} c)}^{3}$

Step 8.2

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

$8 a^{8} b^{2 + 9} c^{12} {(- 3)}^{2} {(- b^{3} c)}^{3}$

Step 8.3

Add $2$ and $9$ .

$8 a^{8} b^{11} c^{12} {(- 3)}^{2} {(- b^{3} c)}^{3}$

Step 9

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

$8 a^{8} b^{11} c^{12} \cdot 9 {(- b^{3} c)}^{3}$

Step 10

Multiply $9$ by $8$ .

$72 a^{8} b^{11} c^{12} {(- b^{3} c)}^{3}$

Step 11

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

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Step 11.1

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

$72 a^{8} b^{11} c^{12} ({(- b^{3})}^{3} c^{3})$

Step 11.2

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

$72 a^{8} b^{11} c^{12} ({(- 1)}^{3} {(b^{3})}^{3} c^{3})$

Step 12

Multiply $c^{12}$ by $c^{3}$ by adding the exponents.

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Step 12.1

Move $c^{3}$ .

$72 a^{8} b^{11} (c^{3} c^{12}) ({(- 1)}^{3} {(b^{3})}^{3})$

Step 12.2

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

$72 a^{8} b^{11} c^{3 + 12} ({(- 1)}^{3} {(b^{3})}^{3})$

Step 12.3

Add $3$ and $12$ .

$72 a^{8} b^{11} c^{15} ({(- 1)}^{3} {(b^{3})}^{3})$

Step 13

Raise $- 1$ to the power of $3$ .

$72 a^{8} b^{11} c^{15} (- {(b^{3})}^{3})$

Step 14

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

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Step 14.1

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

$72 a^{8} b^{11} c^{15} (- b^{3 \cdot 3})$

Step 14.2

Multiply $3$ by $3$ .

$72 a^{8} b^{11} c^{15} (- b^{9})$

Step 15

Multiply $b^{11}$ by $b^{9}$ by adding the exponents.

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Step 15.1

Move $b^{9}$ .

$72 a^{8} (b^{9} b^{11}) c^{15} \cdot - 1$

Step 15.2

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

$72 a^{8} b^{9 + 11} c^{15} \cdot - 1$

Step 15.3

Add $9$ and $11$ .

$72 a^{8} b^{20} c^{15} \cdot - 1$

Step 16

Multiply $- 1$ by $72$ .

$- 72 a^{8} b^{20} c^{15}$