Basic Math Examples

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

Step 1

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $3$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Multiply $3$ by $3$ .

$\frac{a^{6} b^{9}}{a b {(a b)}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $a b$ .

$\frac{a^{6} b^{9}}{a b a^{2} b^{2}}$

Step 3.2

Combine exponents.

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Step 3.2.1

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

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Step 3.2.1.1

Move $a^{2}$ .

$\frac{a^{6} b^{9}}{a^{2} a b \cdot b^{2}}$

Step 3.2.1.2

Multiply $a^{2}$ by $a$ .

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Step 3.2.1.2.1

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

$\frac{a^{6} b^{9}}{a^{2} a^{1} b \cdot b^{2}}$

Step 3.2.1.2.2

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

$\frac{a^{6} b^{9}}{a^{2 + 1} b \cdot b^{2}}$

Step 3.2.1.3

Add $2$ and $1$ .

$\frac{a^{6} b^{9}}{a^{3} b \cdot b^{2}}$

Step 3.2.2

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

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Step 3.2.2.1

Move $b^{2}$ .

$\frac{a^{6} b^{9}}{a^{3} (b^{2} b)}$

Step 3.2.2.2

Multiply $b^{2}$ by $b$ .

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Step 3.2.2.2.1

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

$\frac{a^{6} b^{9}}{a^{3} (b^{2} b^{1})}$

Step 3.2.2.2.2

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

$\frac{a^{6} b^{9}}{a^{3} b^{2 + 1}}$

Step 3.2.2.3

Add $2$ and $1$ .

$\frac{a^{6} b^{9}}{a^{3} b^{3}}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $a^{6}$ and $a^{3}$ .

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Step 4.1.1

Factor $a^{3}$ out of $a^{6} b^{9}$ .

$\frac{a^{3} (a^{3} b^{9})}{a^{3} b^{3}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

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

$\frac{a^{3} (a^{3} b^{9})}{a^{3} (b^{3})}$

Step 4.1.2.2

Cancel the common factor.

$\frac{a^{3} (a^{3} b^{9})}{a^{3} b^{3}}$

Step 4.1.2.3

Rewrite the expression.

$\frac{a^{3} b^{9}}{b^{3}}$

Step 4.2

Cancel the common factor of $b^{9}$ and $b^{3}$ .

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Step 4.2.1

Factor $b^{3}$ out of $a^{3} b^{9}$ .

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Multiply by $1$ .

$\frac{b^{3} (a^{3} b^{6})}{b^{3} \cdot 1}$

Step 4.2.2.2

Cancel the common factor.

$\frac{b^{3} (a^{3} b^{6})}{b^{3} \cdot 1}$

Step 4.2.2.3

Rewrite the expression.

$\frac{a^{3} b^{6}}{1}$

Step 4.2.2.4

Divide $a^{3} b^{6}$ by $1$ .

$a^{3} b^{6}$