Algebra Examples

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

Step 1

Move $c^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $a^{- 1} b^{2} c$ .

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

Step 2.2

Multiply $c^{3}$ by $c^{2}$ by adding the exponents.

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Step 2.2.1

Move $c^{2}$ .

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

Step 2.2.2

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

$\frac{3 a^{2} b^{3}}{{(a^{- 1} b^{2})}^{3} c^{2 + 3}}$

Step 2.2.3

Add $2$ and $3$ .

$\frac{3 a^{2} b^{3}}{{(a^{- 1} b^{2})}^{3} c^{5}}$

Step 2.3

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

$\frac{3 a^{2} b^{3}}{{(\frac{1}{a} b^{2})}^{3} c^{5}}$

Step 2.4

Combine $\frac{1}{a}$ and $b^{2}$ .

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

Step 2.5

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

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

Step 2.6

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

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Step 2.6.1

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

$\frac{3 a^{2} b^{3}}{\frac{b^{2 \cdot 3}}{a^{3}} c^{5}}$

Step 2.6.2

Multiply $2$ by $3$ .

$\frac{3 a^{2} b^{3}}{\frac{b^{6}}{a^{3}} c^{5}}$

Step 3

Combine $\frac{b^{6}}{a^{3}}$ and $c^{5}$ .

$\frac{3 a^{2} b^{3}}{\frac{b^{6} c^{5}}{a^{3}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$3 a^{2} b^{3} \frac{a^{3}}{b^{6} c^{5}}$

Step 5

Cancel the common factor of $b^{3}$ .

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Step 5.1

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

$b^{3} (3 a^{2}) \frac{a^{3}}{b^{6} c^{5}}$

Step 5.2

Factor $b^{3}$ out of $b^{6} c^{5}$ .

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

Step 5.3

Cancel the common factor.

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

Step 5.4

Rewrite the expression.

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

Step 6

Combine $3$ and $\frac{a^{3}}{b^{3} c^{5}}$ .

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

Step 7

Combine $a^{2}$ and $\frac{3 a^{3}}{b^{3} c^{5}}$ .

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

Step 8

Multiply $a^{2}$ by $a^{3}$ by adding the exponents.

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Step 8.1

Move $a^{3}$ .

$\frac{a^{3} a^{2} \cdot 3}{b^{3} c^{5}}$

Step 8.2

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

$\frac{a^{3 + 2} \cdot 3}{b^{3} c^{5}}$

Step 8.3

Add $3$ and $2$ .

$\frac{a^{5} \cdot 3}{b^{3} c^{5}}$

Step 9

Move $3$ to the left of $a^{5}$ .

$\frac{3 a^{5}}{b^{3} c^{5}}$