Basic Math Examples

$\frac{{(9 u^{2} v^{2})}^{- 4} {(4 u^{- 2} v^{3})}^{3}}{{(9 u^{2} v^{- 3})}^{- 2}}$

Step 1

Simplify.

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Step 1.1

Move ${(9 u^{2} v^{2})}^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(4 u^{- 2} v^{3})}^{3}}{{(9 u^{2} v^{- 3})}^{- 2} {(9 u^{2} v^{2})}^{4}}$

Step 1.2

Move ${(9 u^{2} v^{- 3})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(4 u^{- 2} v^{3})}^{3} {(9 u^{2} v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $4 u^{- 2} v^{3}$ .

$\frac{{(4 u^{- 2})}^{3} {(v^{3})}^{3} {(9 u^{2} v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.2

Apply the product rule to $9 u^{2} v^{- 3}$ .

$\frac{{(4 u^{- 2})}^{3} {(v^{3})}^{3} {(9 u^{2})}^{2} {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.3

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

$\frac{{(4 \frac{1}{u^{2}})}^{3} {(v^{3})}^{3} {(9 u^{2})}^{2} {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.4

Combine $4$ and $\frac{1}{u^{2}}$ .

$\frac{{(\frac{4}{u^{2}})}^{3} {(v^{3})}^{3} {(9 u^{2})}^{2} {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.5

Apply the product rule to $\frac{4}{u^{2}}$ .

$\frac{\frac{4^{3}}{{(u^{2})}^{3}} {(v^{3})}^{3} {(9 u^{2})}^{2} {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.6

Raise $4$ to the power of $3$ .

$\frac{\frac{64}{{(u^{2})}^{3}} {(v^{3})}^{3} {(9 u^{2})}^{2} {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.7

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

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Step 2.7.1

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

$\frac{\frac{64}{u^{2 \cdot 3}} {(v^{3})}^{3} {(9 u^{2})}^{2} {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.7.2

Multiply $2$ by $3$ .

$\frac{\frac{64}{u^{6}} {(v^{3})}^{3} {(9 u^{2})}^{2} {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.8

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

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Step 2.8.1

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

$\frac{\frac{64}{u^{6}} v^{3 \cdot 3} {(9 u^{2})}^{2} {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.8.2

Multiply $3$ by $3$ .

$\frac{\frac{64}{u^{6}} v^{9} {(9 u^{2})}^{2} {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.9

Apply the product rule to $9 u^{2}$ .

$\frac{\frac{64}{u^{6}} v^{9} (9^{2} {(u^{2})}^{2}) {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.10

Raise $9$ to the power of $2$ .

$\frac{\frac{64}{u^{6}} v^{9} (81 {(u^{2})}^{2}) {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.11

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

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Step 2.11.1

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

$\frac{\frac{64}{u^{6}} v^{9} (81 u^{2 \cdot 2}) {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.11.2

Multiply $2$ by $2$ .

$\frac{\frac{64}{u^{6}} v^{9} (81 u^{4}) {(v^{- 3})}^{2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.12

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

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Step 2.12.1

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

$\frac{\frac{64}{u^{6}} v^{9} (81 u^{4}) v^{- 3 \cdot 2}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.12.2

Multiply $- 3$ by $2$ .

$\frac{\frac{64}{u^{6}} v^{9} (81 u^{4}) v^{- 6}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.13

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

$\frac{\frac{64}{u^{6}} v^{9} \cdot 81 u^{4} \frac{1}{v^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.14

Combine exponents.

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Step 2.14.1

Combine $\frac{64}{u^{6}}$ and $v^{9}$ .

$\frac{\frac{64 v^{9}}{u^{6}} \cdot 81 u^{4} \frac{1}{v^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.14.2

Combine $\frac{64 v^{9}}{u^{6}}$ and $81$ .

$\frac{\frac{64 v^{9} \cdot 81}{u^{6}} u^{4} \frac{1}{v^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.14.3

Multiply $81$ by $64$ .

$\frac{\frac{5184 v^{9}}{u^{6}} u^{4} \frac{1}{v^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.14.4

Combine $\frac{5184 v^{9}}{u^{6}}$ and $u^{4}$ .

$\frac{\frac{5184 v^{9} u^{4}}{u^{6}} \cdot \frac{1}{v^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.14.5

Multiply $\frac{5184 v^{9} u^{4}}{u^{6}}$ by $\frac{1}{v^{6}}$ .

$\frac{\frac{5184 v^{9} u^{4}}{u^{6} v^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.15

Reduce the expression $\frac{5184 v^{9} u^{4}}{u^{6} v^{6}}$ by cancelling the common factors.

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Step 2.15.1

Factor $v^{6}$ out of $5184 v^{9} u^{4}$ .

$\frac{\frac{v^{6} (5184 v^{3} u^{4})}{u^{6} v^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.15.2

Factor $v^{6}$ out of $u^{6} v^{6}$ .

$\frac{\frac{v^{6} (5184 v^{3} u^{4})}{v^{6} u^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.15.3

Cancel the common factor.

$\frac{\frac{v^{6} (5184 v^{3} u^{4})}{v^{6} u^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.15.4

Rewrite the expression.

$\frac{\frac{5184 v^{3} u^{4}}{u^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.16

Cancel the common factor of $u^{4}$ and $u^{6}$ .

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Step 2.16.1

Factor $u^{4}$ out of $5184 v^{3} u^{4}$ .

$\frac{\frac{u^{4} (5184 v^{3})}{u^{6}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.16.2

Cancel the common factors.

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Step 2.16.2.1

Factor $u^{4}$ out of $u^{6}$ .

$\frac{\frac{u^{4} (5184 v^{3})}{u^{4} u^{2}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.16.2.2

Cancel the common factor.

$\frac{\frac{u^{4} (5184 v^{3})}{u^{4} u^{2}}}{{(9 u^{2} v^{2})}^{4}}$

Step 2.16.2.3

Rewrite the expression.

$\frac{\frac{5184 v^{3}}{u^{2}}}{{(9 u^{2} v^{2})}^{4}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $9 u^{2} v^{2}$ .

$\frac{\frac{5184 v^{3}}{u^{2}}}{{(9 u^{2})}^{4} {(v^{2})}^{4}}$

Step 3.2

Apply the product rule to $9 u^{2}$ .

$\frac{\frac{5184 v^{3}}{u^{2}}}{9^{4} {(u^{2})}^{4} {(v^{2})}^{4}}$

Step 3.3

Raise $9$ to the power of $4$ .

$\frac{\frac{5184 v^{3}}{u^{2}}}{6561 {(u^{2})}^{4} {(v^{2})}^{4}}$

Step 3.4

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

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Step 3.4.1

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

$\frac{\frac{5184 v^{3}}{u^{2}}}{6561 u^{2 \cdot 4} {(v^{2})}^{4}}$

Step 3.4.2

Multiply $2$ by $4$ .

$\frac{\frac{5184 v^{3}}{u^{2}}}{6561 u^{8} {(v^{2})}^{4}}$

Step 3.5

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

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Step 3.5.1

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

$\frac{\frac{5184 v^{3}}{u^{2}}}{6561 u^{8} v^{2 \cdot 4}}$

Step 3.5.2

Multiply $2$ by $4$ .

$\frac{\frac{5184 v^{3}}{u^{2}}}{6561 u^{8} v^{8}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{5184 v^{3}}{u^{2}} \cdot \frac{1}{6561 u^{8} v^{8}}$

Step 5

Combine.

$\frac{5184 v^{3} \cdot 1}{u^{2} (6561 u^{8} v^{8})}$

Step 6

Multiply $u^{2}$ by $u^{8}$ by adding the exponents.

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Step 6.1

Move $u^{8}$ .

$\frac{5184 v^{3} \cdot 1}{u^{8} u^{2} (6561 v^{8})}$

Step 6.2

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

$\frac{5184 v^{3} \cdot 1}{u^{8 + 2} (6561 v^{8})}$

Step 6.3

Add $8$ and $2$ .

$\frac{5184 v^{3} \cdot 1}{u^{10} (6561 v^{8})}$

Step 7

Multiply $5184 v^{3}$ by $1$ .

$\frac{5184 v^{3}}{u^{10} (6561 v^{8})}$

Step 8

Cancel the common factor of $5184$ and $6561$ .

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Step 8.1

Factor $81$ out of $5184 v^{3}$ .

$\frac{81 (64 v^{3})}{u^{10} (6561 v^{8})}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $81$ out of $u^{10} (6561 v^{8})$ .

$\frac{81 (64 v^{3})}{81 (u^{10} (81 v^{8}))}$

Step 8.2.2

Cancel the common factor.

$\frac{81 (64 v^{3})}{81 (u^{10} (81 v^{8}))}$

Step 8.2.3

Rewrite the expression.

$\frac{64 v^{3}}{u^{10} (81 v^{8})}$

Step 9

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

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Step 9.1

Factor $v^{3}$ out of $64 v^{3}$ .

$\frac{v^{3} \cdot 64}{u^{10} (81 v^{8})}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $v^{3}$ out of $u^{10} (81 v^{8})$ .

$\frac{v^{3} \cdot 64}{v^{3} (u^{10} (81 v^{5}))}$

Step 9.2.2

Cancel the common factor.

$\frac{v^{3} \cdot 64}{v^{3} (u^{10} (81 v^{5}))}$

Step 9.2.3

Rewrite the expression.

$\frac{64}{u^{10} (81 v^{5})}$

Step 10

Move $81$ to the left of $u^{10}$ .

$\frac{64}{81 u^{10} v^{5}}$