Pre-Algebra Examples

${(\frac{a^{- 2} b^{4} c^{5}}{a^{- 4} b^{- 4} c^{3}})}^{2}$

Step 1

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

${(\frac{a^{- 2} b^{4} c^{5} a^{4}}{b^{- 4} c^{3}})}^{2}$

Step 2

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

${(\frac{a^{- 2} b^{4} c^{5} a^{4} b^{4}}{c^{3}})}^{2}$

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

Move $a^{4}$ .

${(\frac{a^{4} a^{- 2} b^{4} c^{5} b^{4}}{c^{3}})}^{2}$

Step 3.1.2

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

${(\frac{a^{4 - 2} b^{4} c^{5} b^{4}}{c^{3}})}^{2}$

Step 3.1.3

Subtract $2$ from $4$ .

${(\frac{a^{2} b^{4} c^{5} b^{4}}{c^{3}})}^{2}$

Step 3.2

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

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Step 3.2.1

Move $b^{4}$ .

${(\frac{a^{2} (b^{4} b^{4}) c^{5}}{c^{3}})}^{2}$

Step 3.2.2

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

${(\frac{a^{2} b^{4 + 4} c^{5}}{c^{3}})}^{2}$

Step 3.2.3

Add $4$ and $4$ .

${(\frac{a^{2} b^{8} c^{5}}{c^{3}})}^{2}$

Step 4

Cancel the common factor of $c^{5}$ and $c^{3}$ .

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Step 4.1

Factor $c^{3}$ out of $a^{2} b^{8} c^{5}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.2.4

Divide $a^{2} b^{8} c^{2}$ by $1$ .

${(a^{2} b^{8} c^{2})}^{2}$

Step 5

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

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Step 5.1

Apply the product rule to $a^{2} b^{8} c^{2}$ .

${(a^{2} b^{8})}^{2} {(c^{2})}^{2}$

Step 5.2

Apply the product rule to $a^{2} b^{8}$ .

${(a^{2})}^{2} {(b^{8})}^{2} {(c^{2})}^{2}$

Step 6

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

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Step 6.1

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

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

Step 6.2

Multiply $2$ by $2$ .

$a^{4} {(b^{8})}^{2} {(c^{2})}^{2}$

Step 7

Multiply the exponents in ${(b^{8})}^{2}$ .

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Step 7.1

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

$a^{4} b^{8 \cdot 2} {(c^{2})}^{2}$

Step 7.2

Multiply $8$ by $2$ .

$a^{4} b^{16} {(c^{2})}^{2}$

Step 8

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

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Step 8.1

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

$a^{4} b^{16} c^{2 \cdot 2}$

Step 8.2

Multiply $2$ by $2$ .

$a^{4} b^{16} c^{4}$