Finite Math Examples

$\frac{{(2 n^{6} r^{7})}^{2} {(8 n^{6} r)}^{- 2}}{{(- 3 n r^{9})}^{- 1} {(2 n^{3} r^{9})}^{3}}$

Step 1

Simplify the expression.

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Step 1.1

Move ${(8 n^{6} r)}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{{(2 n^{6} r^{7})}^{2}}{{(- 3 n r^{9})}^{- 1} {(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 1.2

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

$\frac{{(2 n^{6} r^{7})}^{2} (- 3 n r^{9})}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $2 n^{6} r^{7}$ .

$\frac{{(2 n^{6})}^{2} {(r^{7})}^{2} \cdot - 3 n r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.2

Apply the product rule to $2 n^{6}$ .

$\frac{2^{2} {(n^{6})}^{2} {(r^{7})}^{2} \cdot - 3 n r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.3

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

$\frac{4 {(n^{6})}^{2} {(r^{7})}^{2} \cdot - 3 n r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.4

Multiply the exponents in ${(n^{6})}^{2}$ .

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Step 2.4.1

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

$\frac{4 n^{6 \cdot 2} {(r^{7})}^{2} \cdot - 3 n r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.4.2

Multiply $6$ by $2$ .

$\frac{4 n^{12} {(r^{7})}^{2} \cdot - 3 n r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.5

Multiply the exponents in ${(r^{7})}^{2}$ .

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Step 2.5.1

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

$\frac{4 n^{12} r^{7 \cdot 2} \cdot - 3 n r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.5.2

Multiply $7$ by $2$ .

$\frac{4 n^{12} r^{14} \cdot - 3 n r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.6

Combine exponents.

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Step 2.6.1

Multiply $- 3$ by $4$ .

$\frac{- 12 n^{12} r^{14} n r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.6.2

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

$\frac{- 12 (n^{1} n^{12}) r^{14} r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.6.3

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

$\frac{- 12 n^{1 + 12} r^{14} r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.6.4

Add $1$ and $12$ .

$\frac{- 12 n^{13} r^{14} r^{9}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.6.5

Multiply $r^{14}$ by $r^{9}$ by adding the exponents.

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Step 2.6.5.1

Move $r^{9}$ .

$\frac{- 12 n^{13} (r^{9} r^{14})}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.6.5.2

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

$\frac{- 12 n^{13} r^{9 + 14}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 2.6.5.3

Add $9$ and $14$ .

$\frac{- 12 n^{13} r^{23}}{{(2 n^{3} r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $2 n^{3} r^{9}$ .

$\frac{- 12 n^{13} r^{23}}{{(2 n^{3})}^{3} {(r^{9})}^{3} {(8 n^{6} r)}^{2}}$

Step 3.2

Apply the product rule to $8 n^{6} r$ .

$\frac{- 12 n^{13} r^{23}}{{(2 n^{3})}^{3} {(r^{9})}^{3} {(8 n^{6})}^{2} r^{2}}$

Step 3.3

Apply the product rule to $2 n^{3}$ .

$\frac{- 12 n^{13} r^{23}}{2^{3} {(n^{3})}^{3} {(r^{9})}^{3} {(8 n^{6})}^{2} r^{2}}$

Step 3.4

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

$\frac{- 12 n^{13} r^{23}}{8 {(n^{3})}^{3} {(r^{9})}^{3} {(8 n^{6})}^{2} r^{2}}$

Step 3.5

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

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Step 3.5.1

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

$\frac{- 12 n^{13} r^{23}}{8 n^{3 \cdot 3} {(r^{9})}^{3} {(8 n^{6})}^{2} r^{2}}$

Step 3.5.2

Multiply $3$ by $3$ .

$\frac{- 12 n^{13} r^{23}}{8 n^{9} {(r^{9})}^{3} {(8 n^{6})}^{2} r^{2}}$

Step 3.6

Multiply the exponents in ${(r^{9})}^{3}$ .

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Step 3.6.1

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

$\frac{- 12 n^{13} r^{23}}{8 n^{9} r^{9 \cdot 3} {(8 n^{6})}^{2} r^{2}}$

Step 3.6.2

Multiply $9$ by $3$ .

$\frac{- 12 n^{13} r^{23}}{8 n^{9} r^{27} {(8 n^{6})}^{2} r^{2}}$

Step 3.7

Apply the product rule to $8 n^{6}$ .

$\frac{- 12 n^{13} r^{23}}{8 n^{9} r^{27} (8^{2} {(n^{6})}^{2}) r^{2}}$

Step 3.8

Raise $8$ to the power of $2$ .

$\frac{- 12 n^{13} r^{23}}{8 n^{9} r^{27} (64 {(n^{6})}^{2}) r^{2}}$

Step 3.9

Multiply the exponents in ${(n^{6})}^{2}$ .

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Step 3.9.1

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

$\frac{- 12 n^{13} r^{23}}{8 n^{9} r^{27} (64 n^{6 \cdot 2}) r^{2}}$

Step 3.9.2

Multiply $6$ by $2$ .

$\frac{- 12 n^{13} r^{23}}{8 n^{9} r^{27} (64 n^{12}) r^{2}}$

$\frac{- 12 n^{13} r^{23}}{8 n^{9} r^{27} \cdot 64 n^{12} r^{2}}$

Step 3.10

Combine exponents.

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Step 3.10.1

Multiply $64$ by $8$ .

$\frac{- 12 n^{13} r^{23}}{512 n^{9} r^{27} n^{12} r^{2}}$

Step 3.10.2

Multiply $n^{9}$ by $n^{12}$ by adding the exponents.

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Step 3.10.2.1

Move $n^{12}$ .

$\frac{- 12 n^{13} r^{23}}{512 (n^{12} n^{9}) r^{27} r^{2}}$

Step 3.10.2.2

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

$\frac{- 12 n^{13} r^{23}}{512 n^{12 + 9} r^{27} r^{2}}$

Step 3.10.2.3

Add $12$ and $9$ .

$\frac{- 12 n^{13} r^{23}}{512 n^{21} r^{27} r^{2}}$

Step 3.10.3

Multiply $r^{27}$ by $r^{2}$ by adding the exponents.

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Step 3.10.3.1

Move $r^{2}$ .

$\frac{- 12 n^{13} r^{23}}{512 n^{21} (r^{2} r^{27})}$

Step 3.10.3.2

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

$\frac{- 12 n^{13} r^{23}}{512 n^{21} r^{2 + 27}}$

Step 3.10.3.3

Add $2$ and $27$ .

$\frac{- 12 n^{13} r^{23}}{512 n^{21} r^{29}}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $- 12$ and $512$ .

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Step 4.1.1

Factor $4$ out of $- 12 n^{13} r^{23}$ .

$\frac{4 (- 3 n^{13} r^{23})}{512 n^{21} r^{29}}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $4$ out of $512 n^{21} r^{29}$ .

$\frac{4 (- 3 n^{13} r^{23})}{4 (128 n^{21} r^{29})}$

Step 4.1.2.2

Cancel the common factor.

$\frac{4 (- 3 n^{13} r^{23})}{4 (128 n^{21} r^{29})}$

Step 4.1.2.3

Rewrite the expression.

$\frac{- 3 n^{13} r^{23}}{128 n^{21} r^{29}}$

Step 4.2

Cancel the common factor of $n^{13}$ and $n^{21}$ .

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Step 4.2.1

Factor $n^{13}$ out of $- 3 n^{13} r^{23}$ .

$\frac{n^{13} (- 3 r^{23})}{128 n^{21} r^{29}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $n^{13}$ out of $128 n^{21} r^{29}$ .

$\frac{n^{13} (- 3 r^{23})}{n^{13} (128 n^{8} r^{29})}$

Step 4.2.2.2

Cancel the common factor.

$\frac{n^{13} (- 3 r^{23})}{n^{13} (128 n^{8} r^{29})}$

Step 4.2.2.3

Rewrite the expression.

$\frac{- 3 r^{23}}{128 n^{8} r^{29}}$

Step 4.3

Cancel the common factor of $r^{23}$ and $r^{29}$ .

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Step 4.3.1

Factor $r^{23}$ out of $- 3 r^{23}$ .

$\frac{r^{23} \cdot - 3}{128 n^{8} r^{29}}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $r^{23}$ out of $128 n^{8} r^{29}$ .

$\frac{r^{23} \cdot - 3}{r^{23} (128 n^{8} r^{6})}$

Step 4.3.2.2

Cancel the common factor.

$\frac{r^{23} \cdot - 3}{r^{23} (128 n^{8} r^{6})}$

Step 4.3.2.3

Rewrite the expression.

$\frac{- 3}{128 n^{8} r^{6}}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{3}{128 n^{8} r^{6}}$