Algebra Examples

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

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 $- a^{2} b$ .

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

Step 1.2

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

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

Step 2

Rewrite using the commutative property of multiplication.

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

Step 3

Raise $- 1$ to the power of $4$ .

$1 a^{3} ({(a^{2})}^{4} b^{4})$

Step 4

Multiply $a^{3}$ by $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Multiply $2$ by $4$ .

$a^{3} (a^{8} b^{4})$

Step 6

Multiply $a^{3}$ by $a^{8}$ by adding the exponents.

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Step 6.1

Move $a^{8}$ .

$a^{8} a^{3} b^{4}$

Step 6.2

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

$a^{8 + 3} b^{4}$

Step 6.3

Add $8$ and $3$ .

$a^{11} b^{4}$