Basic Math Examples

${(8 a^{2} b^{- 3})}^{4} {(3 a b^{- 2})}^{- 3}$

Step 1

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

${(8 a^{2} \frac{1}{b^{3}})}^{4} {(3 a b^{- 2})}^{- 3}$

Step 2

Multiply $8 a^{2} \frac{1}{b^{3}}$ .

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Step 2.1

Combine $8$ and $\frac{1}{b^{3}}$ .

${(a^{2} \frac{8}{b^{3}})}^{4} {(3 a b^{- 2})}^{- 3}$

Step 2.2

Combine $a^{2}$ and $\frac{8}{b^{3}}$ .

${(\frac{a^{2} \cdot 8}{b^{3}})}^{4} {(3 a b^{- 2})}^{- 3}$

Step 3

Move $8$ to the left of $a^{2}$ .

${(\frac{8 \cdot a^{2}}{b^{3}})}^{4} {(3 a b^{- 2})}^{- 3}$

Step 4

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

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Step 4.1

Apply the product rule to $\frac{8 a^{2}}{b^{3}}$ .

$\frac{{(8 a^{2})}^{4}}{{(b^{3})}^{4}} {(3 a b^{- 2})}^{- 3}$

Step 4.2

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

$\frac{8^{4} {(a^{2})}^{4}}{{(b^{3})}^{4}} {(3 a b^{- 2})}^{- 3}$

Step 5

Simplify the numerator.

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Step 5.1

Raise $8$ to the power of $4$ .

$\frac{4096 {(a^{2})}^{4}}{{(b^{3})}^{4}} {(3 a b^{- 2})}^{- 3}$

Step 5.2

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

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Step 5.2.1

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

$\frac{4096 a^{2 \cdot 4}}{{(b^{3})}^{4}} {(3 a b^{- 2})}^{- 3}$

Step 5.2.2

Multiply $2$ by $4$ .

$\frac{4096 a^{8}}{{(b^{3})}^{4}} {(3 a b^{- 2})}^{- 3}$

Step 6

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

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Step 6.1

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

$\frac{4096 a^{8}}{b^{3 \cdot 4}} {(3 a b^{- 2})}^{- 3}$

Step 6.2

Multiply $3$ by $4$ .

$\frac{4096 a^{8}}{b^{12}} {(3 a b^{- 2})}^{- 3}$

Step 7

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

$\frac{4096 a^{8}}{b^{12}} {(3 a \frac{1}{b^{2}})}^{- 3}$

Step 8

Multiply $3 a \frac{1}{b^{2}}$ .

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Step 8.1

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

$\frac{4096 a^{8}}{b^{12}} {(a \frac{3}{b^{2}})}^{- 3}$

Step 8.2

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

$\frac{4096 a^{8}}{b^{12}} {(\frac{a \cdot 3}{b^{2}})}^{- 3}$

Step 9

Move $3$ to the left of $a$ .

$\frac{4096 a^{8}}{b^{12}} {(\frac{3 \cdot a}{b^{2}})}^{- 3}$

Step 10

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{4096 a^{8}}{b^{12}} {(\frac{b^{2}}{3 a})}^{3}$

Step 11

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

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Step 11.1

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

$\frac{4096 a^{8}}{b^{12}} \cdot \frac{{(b^{2})}^{3}}{{(3 a)}^{3}}$

Step 11.2

Apply the product rule to $3 a$ .

$\frac{4096 a^{8}}{b^{12}} \cdot \frac{{(b^{2})}^{3}}{3^{3} a^{3}}$

Step 12

Combine.

$\frac{4096 a^{8} {(b^{2})}^{3}}{b^{12} (3^{3} a^{3})}$

Step 13

Cancel the common factor of $a^{8}$ and $a^{3}$ .

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Step 13.1

Factor $a^{3}$ out of $4096 a^{8} {(b^{2})}^{3}$ .

$\frac{a^{3} (4096 a^{5} {(b^{2})}^{3})}{b^{12} (3^{3} a^{3})}$

Step 13.2

Cancel the common factors.

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Step 13.2.1

Factor $a^{3}$ out of $b^{12} (3^{3} a^{3})$ .

$\frac{a^{3} (4096 a^{5} {(b^{2})}^{3})}{a^{3} (b^{12} \cdot (3^{3}))}$

Step 13.2.2

Cancel the common factor.

$\frac{a^{3} (4096 a^{5} {(b^{2})}^{3})}{a^{3} (b^{12} \cdot (3^{3}))}$

Step 13.2.3

Rewrite the expression.

$\frac{4096 a^{5} {(b^{2})}^{3}}{b^{12} \cdot (3^{3})}$

Step 14

Simplify the expression.

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Step 14.1

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

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Step 14.1.1

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

$\frac{4096 a^{5} b^{2 \cdot 3}}{b^{12} \cdot 3^{3}}$

Step 14.1.2

Multiply $2$ by $3$ .

$\frac{4096 a^{5} b^{6}}{b^{12} \cdot 3^{3}}$

Step 14.2

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

$\frac{4096 a^{5} b^{6}}{b^{12} \cdot 27}$

Step 15

Cancel the common factor of $b^{6}$ and $b^{12}$ .

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Step 15.1

Factor $b^{6}$ out of $4096 a^{5} b^{6}$ .

$\frac{b^{6} (4096 a^{5})}{b^{12} \cdot 27}$

Step 15.2

Cancel the common factors.

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Step 15.2.1

Factor $b^{6}$ out of $b^{12} \cdot 27$ .

$\frac{b^{6} (4096 a^{5})}{b^{6} (b^{6} \cdot 27)}$

Step 15.2.2

Cancel the common factor.

$\frac{b^{6} (4096 a^{5})}{b^{6} (b^{6} \cdot 27)}$

Step 15.2.3

Rewrite the expression.

$\frac{4096 a^{5}}{b^{6} \cdot 27}$

Step 16

Move $27$ to the left of $b^{6}$ .

$\frac{4096 a^{5}}{27 b^{6}}$