Finite Math Examples

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

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 $3 a^{2} b$ .

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Multiply $2$ by $2$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Apply the product rule to $- 2 a$ .

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

Step 5

Multiply $a^{4}$ by $a^{3}$ by adding the exponents.

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Step 5.1

Move $a^{3}$ .

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

Step 5.2

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

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

Step 5.3

Add $3$ and $4$ .

$9 a^{7} b^{2} ({(- 2)}^{3} {(b^{3})}^{3})$

Step 6

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

$9 a^{7} b^{2} (- 8 {(b^{3})}^{3})$

Step 7

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

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Step 7.1

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

$9 a^{7} b^{2} (- 8 b^{3 \cdot 3})$

Step 7.2

Multiply $3$ by $3$ .

$9 a^{7} b^{2} (- 8 b^{9})$

Step 8

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

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Step 8.1

Move $b^{9}$ .

$9 a^{7} (b^{9} b^{2}) \cdot - 8$

Step 8.2

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

$9 a^{7} b^{9 + 2} \cdot - 8$

Step 8.3

Add $9$ and $2$ .

$9 a^{7} b^{11} \cdot - 8$

Step 9

Multiply $- 8$ by $9$ .

$- 72 a^{7} b^{11}$