Basic Math Examples

$\frac{{(3 a b^{3})}^{2}}{12 a^{2} b^{5}} \cdot \frac{18 a^{4} b}{{(a^{- 1} b)}^{3}}$

Step 1

Simplify terms.

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Step 1.1

Combine.

$\frac{{(3 a b^{3})}^{2} (18 a^{4} b)}{12 a^{2} b^{5} {(a^{- 1} b)}^{3}}$

Step 1.2

Cancel the common factor of $18$ and $12$ .

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Step 1.2.1

Factor $6$ out of ${(3 a b^{3})}^{2} (18 a^{4} b)$ .

$\frac{6 ({(3 a b^{3})}^{2} (3 a^{4} b))}{12 a^{2} b^{5} {(a^{- 1} b)}^{3}}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $6$ out of $12 a^{2} b^{5} {(a^{- 1} b)}^{3}$ .

$\frac{6 ({(3 a b^{3})}^{2} (3 a^{4} b))}{6 (2 a^{2} b^{5} {(a^{- 1} b)}^{3})}$

Step 1.2.2.2

Cancel the common factor.

$\frac{6 ({(3 a b^{3})}^{2} (3 a^{4} b))}{6 (2 a^{2} b^{5} {(a^{- 1} b)}^{3})}$

Step 1.2.2.3

Rewrite the expression.

$\frac{{(3 a b^{3})}^{2} (3 a^{4} b)}{2 a^{2} b^{5} {(a^{- 1} b)}^{3}}$

Step 1.3

Cancel the common factor of $a^{4}$ and $a^{2}$ .

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Step 1.3.1

Factor $a^{2}$ out of ${(3 a b^{3})}^{2} (3 a^{4} b)$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $a^{2}$ out of $2 a^{2} b^{5} {(a^{- 1} b)}^{3}$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $b$ out of $2 b^{5} {(a^{- 1} b)}^{3}$ .

$\frac{b ({(3 a b^{3})}^{2} (3 a^{2}))}{b (2 b^{4} {(a^{- 1} b)}^{3})}$

Step 1.4.2.2

Cancel the common factor.

$\frac{b ({(3 a b^{3})}^{2} (3 a^{2}))}{b (2 b^{4} {(a^{- 1} b)}^{3})}$

Step 1.4.2.3

Rewrite the expression.

$\frac{{(3 a b^{3})}^{2} (3 a^{2})}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(3 a)}^{2} {(b^{3})}^{2} \cdot 3 a^{2}}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 2.2

Apply the product rule to $3 a$ .

$\frac{3^{2} a^{2} {(b^{3})}^{2} \cdot 3 a^{2}}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 2.3

Raise $3$ to the power of $2$ .

$\frac{9 a^{2} {(b^{3})}^{2} \cdot 3 a^{2}}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 2.4

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

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Step 2.4.1

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

$\frac{9 a^{2} b^{3 \cdot 2} \cdot 3 a^{2}}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 2.4.2

Multiply $3$ by $2$ .

$\frac{9 a^{2} b^{6} \cdot 3 a^{2}}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 2.5

Combine exponents.

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Step 2.5.1

Multiply $3$ by $9$ .

$\frac{27 a^{2} b^{6} a^{2}}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 2.5.2

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

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Step 2.5.2.1

Move $a^{2}$ .

$\frac{27 (a^{2} a^{2}) b^{6}}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 2.5.2.2

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

$\frac{27 a^{2 + 2} b^{6}}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 2.5.2.3

Add $2$ and $2$ .

$\frac{27 a^{4} b^{6}}{2 b^{4} {(a^{- 1} b)}^{3}}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{27 a^{4} b^{6}}{2 b^{4} {(a^{- 1})}^{3} b^{3}}$

Step 3.2

Multiply $b^{4}$ by $b^{3}$ by adding the exponents.

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Step 3.2.1

Move $b^{3}$ .

$\frac{27 a^{4} b^{6}}{2 (b^{3} b^{4}) {(a^{- 1})}^{3}}$

Step 3.2.2

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

$\frac{27 a^{4} b^{6}}{2 b^{3 + 4} {(a^{- 1})}^{3}}$

Step 3.2.3

Add $3$ and $4$ .

$\frac{27 a^{4} b^{6}}{2 b^{7} {(a^{- 1})}^{3}}$

Step 3.3

Multiply the exponents in ${(a^{- 1})}^{3}$ .

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Step 3.3.1

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

$\frac{27 a^{4} b^{6}}{2 b^{7} a^{- 1 \cdot 3}}$

Step 3.3.2

Multiply $- 1$ by $3$ .

$\frac{27 a^{4} b^{6}}{2 b^{7} a^{- 3}}$

Step 3.4

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

$\frac{27 a^{4} b^{6}}{2 b^{7} \frac{1}{a^{3}}}$

Step 3.5

Combine exponents.

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Step 3.5.1

Combine $2$ and $\frac{1}{a^{3}}$ .

$\frac{27 a^{4} b^{6}}{b^{7} \frac{2}{a^{3}}}$

Step 3.5.2

Combine $b^{7}$ and $\frac{2}{a^{3}}$ .

$\frac{27 a^{4} b^{6}}{\frac{b^{7} \cdot 2}{a^{3}}}$

Step 3.6

Move $2$ to the left of $b^{7}$ .

$\frac{27 a^{4} b^{6}}{\frac{2 b^{7}}{a^{3}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$27 a^{4} b^{6} \frac{a^{3}}{2 b^{7}}$

Step 5

Cancel the common factor of $b^{6}$ .

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Step 5.1

Factor $b^{6}$ out of $27 a^{4} b^{6}$ .

$b^{6} (27 a^{4}) \frac{a^{3}}{2 b^{7}}$

Step 5.2

Factor $b^{6}$ out of $2 b^{7}$ .

$b^{6} (27 a^{4}) \frac{a^{3}}{b^{6} (2 b)}$

Step 5.3

Cancel the common factor.

$b^{6} (27 a^{4}) \frac{a^{3}}{b^{6} (2 b)}$

Step 5.4

Rewrite the expression.

$27 a^{4} \frac{a^{3}}{2 b}$

Step 6

Combine $27$ and $\frac{a^{3}}{2 b}$ .

$a^{4} \frac{27 a^{3}}{2 b}$

Step 7

Combine $a^{4}$ and $\frac{27 a^{3}}{2 b}$ .

$\frac{a^{4} (27 a^{3})}{2 b}$

Step 8

Multiply $a^{4}$ by $a^{3}$ by adding the exponents.

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Step 8.1

Move $a^{3}$ .

$\frac{a^{3} a^{4} \cdot 27}{2 b}$

Step 8.2

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

$\frac{a^{3 + 4} \cdot 27}{2 b}$

Step 8.3

Add $3$ and $4$ .

$\frac{a^{7} \cdot 27}{2 b}$

Step 9

Move $27$ to the left of $a^{7}$ .

$\frac{27 a^{7}}{2 b}$