Algebra Examples

${(- 3 p^{- 3} q \cdot 3 p^{3} q^{- 4})}^{2}$

Step 1

Multiply $p^{- 3}$ by $p^{3}$ by adding the exponents.

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Step 1.1

Move $p^{3}$ .

${(- 3 (p^{3} p^{- 3}) q \cdot (3 q^{- 4}))}^{2}$

Step 1.2

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

${(- 3 p^{3 - 3} q \cdot (3 q^{- 4}))}^{2}$

Step 1.3

Subtract $3$ from $3$ .

${(- 3 p^{0} q \cdot (3 q^{- 4}))}^{2}$

Step 2

Simplify $- 3 p^{0} q \cdot (3 q^{- 4})$ .

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

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

Multiply $q$ by $q^{- 4}$ by adding the exponents.

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Step 4.1

Move $q^{- 4}$ .

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

Step 4.2

Multiply $q^{- 4}$ by $q$ .

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Step 4.2.1

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

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

Step 4.2.2

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

${(- 3 \cdot 3 q^{- 4 + 1})}^{2}$

Step 4.3

Add $- 4$ and $1$ .

${(- 3 \cdot 3 q^{- 3})}^{2}$

Step 5

Multiply $- 3$ by $3$ .

${(- 9 q^{- 3})}^{2}$

Step 6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(- 9 \frac{1}{q^{3}})}^{2}$

Step 7

Combine $- 9$ and $\frac{1}{q^{3}}$ .

${(\frac{- 9}{q^{3}})}^{2}$

Step 8

Move the negative in front of the fraction.

${(- \frac{9}{q^{3}})}^{2}$

Step 9

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

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Step 9.1

Apply the product rule to $- \frac{9}{q^{3}}$ .

${(- 1)}^{2} {(\frac{9}{q^{3}})}^{2}$

Step 9.2

Apply the product rule to $\frac{9}{q^{3}}$ .

${(- 1)}^{2} \frac{9^{2}}{{(q^{3})}^{2}}$

Step 10

Raise $- 1$ to the power of $2$ .

$1 \frac{9^{2}}{{(q^{3})}^{2}}$

Step 11

Multiply $\frac{9^{2}}{{(q^{3})}^{2}}$ by $1$ .

$\frac{9^{2}}{{(q^{3})}^{2}}$

Step 12

Raise $9$ to the power of $2$ .

$\frac{81}{{(q^{3})}^{2}}$

Step 13

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

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Step 13.1

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

$\frac{81}{q^{3 \cdot 2}}$

Step 13.2

Multiply $3$ by $2$ .

$\frac{81}{q^{6}}$