Basic Math Examples

$\frac{{(- 3 p q)}^{- 2} \cdot {(- 2 p^{2} q^{- 1} r)}^{- 3}}{{(- 6 p^{2} q)}^{- 4}}$

Step 1

Simplify the expression.

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Step 1.1

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

$\frac{{(- 2 p^{2} q^{- 1} r)}^{- 3}}{{(- 6 p^{2} q)}^{- 4} {(- 3 p q)}^{2}}$

Step 1.2

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

$\frac{1}{{(- 6 p^{2} q)}^{- 4} {(- 3 p q)}^{2} {(- 2 p^{2} q^{- 1} r)}^{3}}$

Step 1.3

Move ${(- 6 p^{2} q)}^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(- 6 p^{2} q)}^{4}}{{(- 3 p q)}^{2} {(- 2 p^{2} q^{- 1} r)}^{3}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $- 6 p^{2} q$ .

$\frac{{(- 6 p^{2})}^{4} q^{4}}{{(- 3 p q)}^{2} {(- 2 p^{2} q^{- 1} r)}^{3}}$

Step 2.2

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

$\frac{{(- 6)}^{4} {(p^{2})}^{4} q^{4}}{{(- 3 p q)}^{2} {(- 2 p^{2} q^{- 1} r)}^{3}}$

Step 2.3

Raise $- 6$ to the power of $4$ .

$\frac{1296 {(p^{2})}^{4} q^{4}}{{(- 3 p q)}^{2} {(- 2 p^{2} q^{- 1} r)}^{3}}$

Step 2.4

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

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Step 2.4.1

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

$\frac{1296 p^{2 \cdot 4} q^{4}}{{(- 3 p q)}^{2} {(- 2 p^{2} q^{- 1} r)}^{3}}$

Step 2.4.2

Multiply $2$ by $4$ .

$\frac{1296 p^{8} q^{4}}{{(- 3 p q)}^{2} {(- 2 p^{2} q^{- 1} r)}^{3}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $- 3 p q$ .

$\frac{1296 p^{8} q^{4}}{{(- 3 p)}^{2} q^{2} {(- 2 p^{2} q^{- 1} r)}^{3}}$

Step 3.2

Apply the product rule to $- 2 p^{2} q^{- 1} r$ .

$\frac{1296 p^{8} q^{4}}{{(- 3 p)}^{2} q^{2} {(- 2 p^{2} q^{- 1})}^{3} r^{3}}$

Step 3.3

Apply the product rule to $- 3 p$ .

$\frac{1296 p^{8} q^{4}}{{(- 3)}^{2} p^{2} q^{2} {(- 2 p^{2} q^{- 1})}^{3} r^{3}}$

Step 3.4

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

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} {(- 2 p^{2} q^{- 1})}^{3} r^{3}}$

Step 3.5

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

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} {(- 2 p^{2} \frac{1}{q})}^{3} r^{3}}$

Step 3.6

Multiply $- 2 p^{2} \frac{1}{q}$ .

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Step 3.6.1

Combine $- 2$ and $\frac{1}{q}$ .

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} {(p^{2} \frac{- 2}{q})}^{3} r^{3}}$

Step 3.6.2

Combine $p^{2}$ and $\frac{- 2}{q}$ .

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} {(\frac{p^{2} \cdot - 2}{q})}^{3} r^{3}}$

Step 3.7

Move $- 2$ to the left of $p^{2}$ .

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} {(\frac{- 2 \cdot p^{2}}{q})}^{3} r^{3}}$

Step 3.8

Move the negative in front of the fraction.

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} {(- \frac{2 \cdot p^{2}}{q})}^{3} r^{3}}$

Step 3.9

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

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Step 3.9.1

Apply the product rule to $- \frac{2 p^{2}}{q}$ .

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} ({(- 1)}^{3} {(\frac{2 p^{2}}{q})}^{3}) r^{3}}$

Step 3.9.2

Apply the product rule to $\frac{2 p^{2}}{q}$ .

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} ({(- 1)}^{3} \frac{{(2 p^{2})}^{3}}{q^{3}}) r^{3}}$

Step 3.9.3

Apply the product rule to $2 p^{2}$ .

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} ({(- 1)}^{3} \frac{2^{3} {(p^{2})}^{3}}{q^{3}}) r^{3}}$

Step 3.10

Raise $- 1$ to the power of $3$ .

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} (- \frac{2^{3} {(p^{2})}^{3}}{q^{3}}) r^{3}}$

Step 3.11

Simplify the numerator.

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Step 3.11.1

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

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} (- \frac{8 {(p^{2})}^{3}}{q^{3}}) r^{3}}$

Step 3.11.2

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

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Step 3.11.2.1

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

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} (- \frac{8 p^{2 \cdot 3}}{q^{3}}) r^{3}}$

Step 3.11.2.2

Multiply $2$ by $3$ .

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} (- \frac{8 p^{6}}{q^{3}}) r^{3}}$

$\frac{1296 p^{8} q^{4}}{9 p^{2} q^{2} \cdot - 1 \frac{8 p^{6}}{q^{3}} r^{3}}$

Step 3.12

Combine exponents.

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Step 3.12.1

Factor out negative.

$\frac{1296 p^{8} q^{4}}{- (9 p^{2} q^{2} \frac{8 p^{6}}{q^{3}} r^{3})}$

Step 3.12.2

Combine $9$ and $\frac{8 p^{6}}{q^{3}}$ .

$\frac{1296 p^{8} q^{4}}{- (p^{2} q^{2} \frac{9 (8 p^{6})}{q^{3}} r^{3})}$

Step 3.12.3

Multiply $8$ by $9$ .

$\frac{1296 p^{8} q^{4}}{- (p^{2} q^{2} \frac{72 p^{6}}{q^{3}} r^{3})}$

Step 3.12.4

Combine $p^{2}$ and $\frac{72 p^{6}}{q^{3}}$ .

$\frac{1296 p^{8} q^{4}}{- (q^{2} \frac{p^{2} (72 p^{6})}{q^{3}} r^{3})}$

Step 3.12.5

Multiply $p^{2}$ by $p^{6}$ by adding the exponents.

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Step 3.12.5.1

Move $p^{6}$ .

$\frac{1296 p^{8} q^{4}}{- (q^{2} \frac{p^{6} p^{2} \cdot 72}{q^{3}} r^{3})}$

Step 3.12.5.2

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

$\frac{1296 p^{8} q^{4}}{- (q^{2} \frac{p^{6 + 2} \cdot 72}{q^{3}} r^{3})}$

Step 3.12.5.3

Add $6$ and $2$ .

$\frac{1296 p^{8} q^{4}}{- (q^{2} \frac{p^{8} \cdot 72}{q^{3}} r^{3})}$

Step 3.12.6

Combine $q^{2}$ and $\frac{p^{8} \cdot 72}{q^{3}}$ .

$\frac{1296 p^{8} q^{4}}{- (\frac{q^{2} (p^{8} \cdot 72)}{q^{3}} r^{3})}$

Step 3.12.7

Combine $\frac{q^{2} (p^{8} \cdot 72)}{q^{3}}$ and $r^{3}$ .

$\frac{1296 p^{8} q^{4}}{- \frac{q^{2} (p^{8} \cdot 72) r^{3}}{q^{3}}}$

Step 3.13

Remove unnecessary parentheses.

$\frac{1296 p^{8} q^{4}}{- \frac{q^{2} p^{8} \cdot 72 r^{3}}{q^{3}}}$

Step 3.14

Reduce the expression $\frac{q^{2} p^{8} \cdot 72 r^{3}}{q^{3}}$ by cancelling the common factors.

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Step 3.14.1

Factor $q^{2}$ out of $q^{2} p^{8} \cdot 72 r^{3}$ .

$\frac{1296 p^{8} q^{4}}{- \frac{q^{2} (p^{8} \cdot 72 r^{3})}{q^{3}}}$

Step 3.14.2

Factor $q^{2}$ out of $q^{3}$ .

$\frac{1296 p^{8} q^{4}}{- \frac{q^{2} (p^{8} \cdot 72 r^{3})}{q^{2} q}}$

Step 3.14.3

Cancel the common factor.

$\frac{1296 p^{8} q^{4}}{- \frac{q^{2} (p^{8} \cdot 72 r^{3})}{q^{2} q}}$

Step 3.14.4

Rewrite the expression.

$\frac{1296 p^{8} q^{4}}{- \frac{p^{8} \cdot 72 r^{3}}{q}}$

Step 3.15

Move $72$ to the left of $p^{8}$ .

$\frac{1296 p^{8} q^{4}}{- \frac{72 p^{8} r^{3}}{q}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$1296 p^{8} q^{4} (- \frac{q}{72 p^{8} r^{3}})$

Step 5

Cancel the common factor of $72 p^{8}$ .

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Step 5.1

Move the leading negative in $- \frac{q}{72 p^{8} r^{3}}$ into the numerator.

$1296 p^{8} q^{4} \frac{- q}{72 p^{8} r^{3}}$

Step 5.2

Factor $72 p^{8}$ out of $1296 p^{8} q^{4}$ .

$72 p^{8} (18 q^{4}) \frac{- q}{72 p^{8} r^{3}}$

Step 5.3

Factor $72 p^{8}$ out of $72 p^{8} r^{3}$ .

$72 p^{8} (18 q^{4}) \frac{- q}{72 p^{8} (r^{3})}$

Step 5.4

Cancel the common factor.

$72 p^{8} (18 q^{4}) \frac{- q}{72 p^{8} r^{3}}$

Step 5.5

Rewrite the expression.

$18 q^{4} \frac{- q}{r^{3}}$

Step 6

Combine $18$ and $\frac{- q}{r^{3}}$ .

$q^{4} \frac{18 (- q)}{r^{3}}$

Step 7

Multiply $- 1$ by $18$ .

$q^{4} \frac{- 18 q}{r^{3}}$

Step 8

Combine $q^{4}$ and $\frac{- 18 q}{r^{3}}$ .

$\frac{q^{4} (- 18 q)}{r^{3}}$

Step 9

Multiply $q^{4}$ by $q$ by adding the exponents.

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Step 9.1

Move $q$ .

$\frac{q \cdot q^{4} \cdot - 18}{r^{3}}$

Step 9.2

Multiply $q$ by $q^{4}$ .

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Step 9.2.1

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

$\frac{q^{1} q^{4} \cdot - 18}{r^{3}}$

Step 9.2.2

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

$\frac{q^{1 + 4} \cdot - 18}{r^{3}}$

Step 9.3

Add $1$ and $4$ .

$\frac{q^{5} \cdot - 18}{r^{3}}$

Step 10

Move $- 18$ to the left of $q^{5}$ .

$\frac{- 18 \cdot q^{5}}{r^{3}}$

Step 11

Move the negative in front of the fraction.

$- \frac{18 q^{5}}{r^{3}}$