Precalculus Examples

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

Step 1

Move $c^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Move $b^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 3

Multiply $b^{2}$ by $b^{4}$ by adding the exponents.

Tap for more steps...

Step 3.1

Move $b^{4}$ .

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

Step 3.2

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

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

Step 3.3

Add $4$ and $2$ .

${(\frac{a b^{6}}{2 a^{3} c^{3}})}^{- 2}$

Step 4

Cancel the common factor of $a$ and $a^{3}$ .

Tap for more steps...

Step 4.1

Factor $a$ out of $a b^{6}$ .

${(\frac{a (b^{6})}{2 a^{3} c^{3}})}^{- 2}$

Step 4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1

Factor $a$ out of $2 a^{3} c^{3}$ .

${(\frac{a (b^{6})}{a (2 a^{2} c^{3})})}^{- 2}$

Step 4.2.2

Cancel the common factor.

${(\frac{a b^{6}}{a (2 a^{2} c^{3})})}^{- 2}$

Step 4.2.3

Rewrite the expression.

${(\frac{b^{6}}{2 a^{2} c^{3}})}^{- 2}$

Step 5

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{2 a^{2} c^{3}}{b^{6}})}^{2}$

Step 6

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

Tap for more steps...

Step 6.1

Apply the product rule to $\frac{2 a^{2} c^{3}}{b^{6}}$ .

$\frac{{(2 a^{2} c^{3})}^{2}}{{(b^{6})}^{2}}$

Step 6.2

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

$\frac{{(2 a^{2})}^{2} {(c^{3})}^{2}}{{(b^{6})}^{2}}$

Step 6.3

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

$\frac{2^{2} {(a^{2})}^{2} {(c^{3})}^{2}}{{(b^{6})}^{2}}$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

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

$\frac{4 {(a^{2})}^{2} {(c^{3})}^{2}}{{(b^{6})}^{2}}$

Step 7.2

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

Tap for more steps...

Step 7.2.1

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

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

Step 7.2.2

Multiply $2$ by $2$ .

$\frac{4 a^{4} {(c^{3})}^{2}}{{(b^{6})}^{2}}$

Step 7.3

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

Tap for more steps...

Step 7.3.1

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

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

Step 7.3.2

Multiply $3$ by $2$ .

$\frac{4 a^{4} c^{6}}{{(b^{6})}^{2}}$

Step 8

Multiply the exponents in ${(b^{6})}^{2}$ .

Tap for more steps...

Step 8.1

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

$\frac{4 a^{4} c^{6}}{b^{6 \cdot 2}}$

Step 8.2

Multiply $6$ by $2$ .

$\frac{4 a^{4} c^{6}}{b^{12}}$