Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $4 a^{- 3} b^{2}$ .

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

Step 1.2

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

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

Step 1.3

Combine $4$ and $\frac{1}{a^{3}}$ .

$\frac{{(\frac{4}{a^{3}})}^{4} {(b^{2})}^{4} b}{{(12 a^{2} b)}^{3}}$

Step 1.4

Apply the product rule to $\frac{4}{a^{3}}$ .

$\frac{\frac{4^{4}}{{(a^{3})}^{4}} {(b^{2})}^{4} b}{{(12 a^{2} b)}^{3}}$

Step 1.5

Raise $4$ to the power of $4$ .

$\frac{\frac{256}{{(a^{3})}^{4}} {(b^{2})}^{4} b}{{(12 a^{2} b)}^{3}}$

Step 1.6

Multiply the exponents in ${(a^{3})}^{4}$ .

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Step 1.6.1

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

$\frac{\frac{256}{a^{3 \cdot 4}} {(b^{2})}^{4} b}{{(12 a^{2} b)}^{3}}$

Step 1.6.2

Multiply $3$ by $4$ .

$\frac{\frac{256}{a^{12}} {(b^{2})}^{4} b}{{(12 a^{2} b)}^{3}}$

Step 1.7

Multiply the exponents in ${(b^{2})}^{4}$ .

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Step 1.7.1

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

$\frac{\frac{256}{a^{12}} b^{2 \cdot 4} b}{{(12 a^{2} b)}^{3}}$

Step 1.7.2

Multiply $2$ by $4$ .

$\frac{\frac{256}{a^{12}} b^{8} b}{{(12 a^{2} b)}^{3}}$

Step 1.8

Combine exponents.

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Step 1.8.1

Combine $\frac{256}{a^{12}}$ and $b^{8}$ .

$\frac{\frac{256 b^{8}}{a^{12}} b}{{(12 a^{2} b)}^{3}}$

Step 1.8.2

Combine $\frac{256 b^{8}}{a^{12}}$ and $b$ .

$\frac{\frac{256 b^{8} b}{a^{12}}}{{(12 a^{2} b)}^{3}}$

Step 1.8.3

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

$\frac{\frac{256 (b^{1} b^{8})}{a^{12}}}{{(12 a^{2} b)}^{3}}$

Step 1.8.4

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

$\frac{\frac{256 b^{1 + 8}}{a^{12}}}{{(12 a^{2} b)}^{3}}$

Step 1.8.5

Add $1$ and $8$ .

$\frac{\frac{256 b^{9}}{a^{12}}}{{(12 a^{2} b)}^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $12 a^{2} b$ .

$\frac{\frac{256 b^{9}}{a^{12}}}{{(12 a^{2})}^{3} b^{3}}$

Step 2.2

Apply the product rule to $12 a^{2}$ .

$\frac{\frac{256 b^{9}}{a^{12}}}{12^{3} {(a^{2})}^{3} b^{3}}$

Step 2.3

Raise $12$ to the power of $3$ .

$\frac{\frac{256 b^{9}}{a^{12}}}{1728 {(a^{2})}^{3} b^{3}}$

Step 2.4

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

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Step 2.4.1

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

$\frac{\frac{256 b^{9}}{a^{12}}}{1728 a^{2 \cdot 3} b^{3}}$

Step 2.4.2

Multiply $2$ by $3$ .

$\frac{\frac{256 b^{9}}{a^{12}}}{1728 a^{6} b^{3}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{256 b^{9}}{a^{12}} \cdot \frac{1}{1728 a^{6} b^{3}}$

Step 4

Combine.

$\frac{256 b^{9} \cdot 1}{a^{12} (1728 a^{6} b^{3})}$

Step 5

Multiply $a^{12}$ by $a^{6}$ by adding the exponents.

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Step 5.1

Move $a^{6}$ .

$\frac{256 b^{9} \cdot 1}{a^{6} a^{12} (1728 b^{3})}$

Step 5.2

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

$\frac{256 b^{9} \cdot 1}{a^{6 + 12} (1728 b^{3})}$

Step 5.3

Add $6$ and $12$ .

$\frac{256 b^{9} \cdot 1}{a^{18} (1728 b^{3})}$

Step 6

Multiply $256 b^{9}$ by $1$ .

$\frac{256 b^{9}}{a^{18} (1728 b^{3})}$

Step 7

Cancel the common factor of $256$ and $1728$ .

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Step 7.1

Factor $64$ out of $256 b^{9}$ .

$\frac{64 (4 b^{9})}{a^{18} (1728 b^{3})}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $64$ out of $a^{18} (1728 b^{3})$ .

$\frac{64 (4 b^{9})}{64 (a^{18} (27 b^{3}))}$

Step 7.2.2

Cancel the common factor.

$\frac{64 (4 b^{9})}{64 (a^{18} (27 b^{3}))}$

Step 7.2.3

Rewrite the expression.

$\frac{4 b^{9}}{a^{18} (27 b^{3})}$

Step 8

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

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Step 8.1

Factor $b^{3}$ out of $4 b^{9}$ .

$\frac{b^{3} (4 b^{6})}{a^{18} (27 b^{3})}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $b^{3}$ out of $a^{18} (27 b^{3})$ .

$\frac{b^{3} (4 b^{6})}{b^{3} (a^{18} \cdot (27))}$

Step 8.2.2

Cancel the common factor.

$\frac{b^{3} (4 b^{6})}{b^{3} (a^{18} \cdot (27))}$

Step 8.2.3

Rewrite the expression.

$\frac{4 b^{6}}{a^{18} \cdot (27)}$

Step 9

Move $27$ to the left of $a^{18}$ .

$\frac{4 b^{6}}{27 a^{18}}$