Algebra Examples

${(- 4 a^{3} b^{2})}^{2} \cdot \frac{1}{2} a^{3} b$

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 $- 4 a^{3} b^{2}$ .

${(- 4 a^{3})}^{2} {(b^{2})}^{2} \cdot \frac{1}{2} \cdot (a^{3} b)$

Step 1.2

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

${(- 4)}^{2} {(a^{3})}^{2} {(b^{2})}^{2} \cdot \frac{1}{2} \cdot (a^{3} b)$

Step 2

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

$16 {(a^{3})}^{2} {(b^{2})}^{2} \cdot \frac{1}{2} \cdot (a^{3} b)$

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Factor $2$ out of $16 {(a^{3})}^{2} {(b^{2})}^{2}$ .

$2 (8 {(a^{3})}^{2} {(b^{2})}^{2}) \cdot \frac{1}{2} \cdot (a^{3} b)$

Step 3.2

Cancel the common factor.

$2 (8 {(a^{3})}^{2} {(b^{2})}^{2}) \cdot \frac{1}{2} \cdot (a^{3} b)$

Step 3.3

Rewrite the expression.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $3$ by $2$ .

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

Step 5

Multiply $a^{6}$ by $a^{3}$ by adding the exponents.

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Step 5.1

Move $a^{3}$ .

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

Step 5.2

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

$8 a^{3 + 6} {(b^{2})}^{2} \cdot b$

Step 5.3

Add $3$ and $6$ .

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

Step 6

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

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Step 6.1

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

$8 a^{9} b^{2 \cdot 2} \cdot b$

Step 6.2

Multiply $2$ by $2$ .

$8 a^{9} b^{4} \cdot b$

Step 7

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

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Step 7.1

Move $b$ .

$8 a^{9} (b \cdot b^{4})$

Step 7.2

Multiply $b$ by $b^{4}$ .

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Step 7.2.1

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

$8 a^{9} (b^{1} b^{4})$

Step 7.2.2

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

$8 a^{9} b^{1 + 4}$

Step 7.3

Add $1$ and $4$ .

$8 a^{9} b^{5}$