Basic Math Examples

$\frac{{(21 \cdot a \cdot b^{2})}^{- 3}}{{(14^{2} \cdot a \cdot b^{- 2})}^{- 3} \cdot 3^{- 2} \cdot a^{3} \cdot b^{- 2}}$

Step 1

Simplify the expression.

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Step 1.1

Move ${(21 \cdot a \cdot b^{2})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(14^{2} \cdot a \cdot b^{- 2})}^{- 3} \cdot 3^{- 2} \cdot a^{3} \cdot b^{- 2} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 1.2

Move ${(14^{2} \cdot a \cdot b^{- 2})}^{- 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(14^{2} \cdot a \cdot b^{- 2})}^{3}}{3^{- 2} \cdot a^{3} \cdot b^{- 2} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 1.3

Move $3^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(14^{2} \cdot a \cdot b^{- 2})}^{3} \cdot 3^{2}}{a^{3} \cdot b^{- 2} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 1.4

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

$\frac{{(14^{2} \cdot a \cdot b^{- 2})}^{3} \cdot 3^{2} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $14^{2} \cdot a \cdot b^{- 2}$ .

$\frac{{(14^{2} \cdot a)}^{3} \cdot {(b^{- 2})}^{3} \cdot 3^{2} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.2

Raise $14$ to the power of $2$ .

$\frac{{(196 \cdot a)}^{3} \cdot {(b^{- 2})}^{3} \cdot 3^{2} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.3

Apply the product rule to $196 a$ .

$\frac{196^{3} a^{3} \cdot {(b^{- 2})}^{3} \cdot 3^{2} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.4

Raise $196$ to the power of $3$ .

$\frac{7529536 a^{3} \cdot {(b^{- 2})}^{3} \cdot 3^{2} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{7529536 a^{3} \cdot b^{- 2 \cdot 3} \cdot 3^{2} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.5.2

Multiply $- 2$ by $3$ .

$\frac{7529536 a^{3} \cdot b^{- 6} \cdot 3^{2} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.6

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

$\frac{7529536 a^{3} \cdot \frac{1}{b^{6}} \cdot 3^{2} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.7

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

$\frac{7529536 a^{3} \cdot \frac{1}{b^{6}} \cdot 9 b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.8

Combine exponents.

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Step 2.8.1

Combine $7529536$ and $\frac{1}{b^{6}}$ .

$\frac{a^{3} \cdot \frac{7529536}{b^{6}} \cdot 9 b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.8.2

Combine $a^{3}$ and $\frac{7529536}{b^{6}}$ .

$\frac{\frac{a^{3} \cdot 7529536}{b^{6}} \cdot 9 b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.8.3

Combine $\frac{a^{3} \cdot 7529536}{b^{6}}$ and $9$ .

$\frac{\frac{a^{3} \cdot 7529536 \cdot 9}{b^{6}} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.8.4

Multiply $9$ by $7529536$ .

$\frac{\frac{a^{3} \cdot 67765824}{b^{6}} b^{2}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.8.5

Combine $\frac{a^{3} \cdot 67765824}{b^{6}}$ and $b^{2}$ .

$\frac{\frac{a^{3} \cdot 67765824 b^{2}}{b^{6}}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.9

Reduce the expression $\frac{a^{3} \cdot 67765824 b^{2}}{b^{6}}$ by cancelling the common factors.

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Step 2.9.1

Factor $b^{2}$ out of $a^{3} \cdot 67765824 b^{2}$ .

$\frac{\frac{b^{2} (a^{3} \cdot 67765824)}{b^{6}}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.9.2

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

$\frac{\frac{b^{2} (a^{3} \cdot 67765824)}{b^{2} b^{4}}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.9.3

Cancel the common factor.

$\frac{\frac{b^{2} (a^{3} \cdot 67765824)}{b^{2} b^{4}}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.9.4

Rewrite the expression.

$\frac{\frac{a^{3} \cdot 67765824}{b^{4}}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 2.10

Move $67765824$ to the left of $a^{3}$ .

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{3} {(21 \cdot a \cdot b^{2})}^{3}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $21 \cdot a \cdot b^{2}$ .

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{3} {(21 \cdot a)}^{3} \cdot {(b^{2})}^{3}}$

Step 3.2

Apply the product rule to $21 a$ .

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{3} (21^{3} a^{3}) \cdot {(b^{2})}^{3}}$

Step 3.3

Raise $21$ to the power of $3$ .

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{3} (9261 a^{3}) \cdot {(b^{2})}^{3}}$

Step 3.4

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

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Step 3.4.1

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

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{3} (9261 a^{3}) \cdot b^{2 \cdot 3}}$

Step 3.4.2

Multiply $2$ by $3$ .

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{3} (9261 a^{3}) \cdot b^{6}}$

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{3} \cdot 9261 a^{3} \cdot b^{6}}$

Step 3.5

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

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Step 3.5.1

Move $a^{3}$ .

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{3} a^{3} \cdot 9261 \cdot b^{6}}$

Step 3.5.2

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

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{3 + 3} \cdot 9261 \cdot b^{6}}$

Step 3.5.3

Add $3$ and $3$ .

$\frac{\frac{67765824 a^{3}}{b^{4}}}{a^{6} \cdot 9261 \cdot b^{6}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{67765824 a^{3}}{b^{4}} \cdot \frac{1}{a^{6} \cdot 9261 \cdot b^{6}}$

Step 5

Combine.

$\frac{67765824 a^{3} \cdot 1}{b^{4} (a^{6} \cdot 9261 \cdot b^{6})}$

Step 6

Multiply $b^{4}$ by $b^{6}$ by adding the exponents.

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Step 6.1

Move $b^{6}$ .

$\frac{67765824 a^{3} \cdot 1}{b^{6} b^{4} (a^{6} \cdot 9261)}$

Step 6.2

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

$\frac{67765824 a^{3} \cdot 1}{b^{6 + 4} (a^{6} \cdot 9261)}$

Step 6.3

Add $6$ and $4$ .

$\frac{67765824 a^{3} \cdot 1}{b^{10} (a^{6} \cdot 9261)}$

Step 7

Multiply $67765824 a^{3}$ by $1$ .

$\frac{67765824 a^{3}}{b^{10} (a^{6} \cdot 9261)}$

Step 8

Cancel the common factor of $67765824$ and $9261$ .

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Step 8.1

Factor $3087$ out of $67765824 a^{3}$ .

$\frac{3087 (21952 a^{3})}{b^{10} (a^{6} \cdot 9261)}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $3087$ out of $b^{10} (a^{6} \cdot 9261)$ .

$\frac{3087 (21952 a^{3})}{3087 (b^{10} (a^{6} \cdot 3))}$

Step 8.2.2

Cancel the common factor.

$\frac{3087 (21952 a^{3})}{3087 (b^{10} (a^{6} \cdot 3))}$

Step 8.2.3

Rewrite the expression.

$\frac{21952 a^{3}}{b^{10} (a^{6} \cdot 3)}$

Step 9

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

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Step 9.1

Factor $a^{3}$ out of $21952 a^{3}$ .

$\frac{a^{3} \cdot 21952}{b^{10} (a^{6} \cdot 3)}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $a^{3}$ out of $b^{10} (a^{6} \cdot 3)$ .

$\frac{a^{3} \cdot 21952}{a^{3} (b^{10} (a^{3} \cdot 3))}$

Step 9.2.2

Cancel the common factor.

$\frac{a^{3} \cdot 21952}{a^{3} (b^{10} (a^{3} \cdot 3))}$

Step 9.2.3

Rewrite the expression.

$\frac{21952}{b^{10} (a^{3} \cdot 3)}$

Step 10

Move $3$ to the left of $b^{10} a^{3}$ .

$\frac{21952}{3 b^{10} a^{3}}$