Basic Math Examples

$\frac{{(- 2 j^{- 3} w^{5})}^{- 3}}{{(4 j^{- 5} w^{- 5})}^{2}}$

Step 1

Move ${(- 2 j^{- 3} w^{5})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(4 j^{- 5} w^{- 5})}^{2} {(- 2 j^{- 3} w^{5})}^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $4 j^{- 5} w^{- 5}$ .

$\frac{1}{{(4 j^{- 5})}^{2} {(w^{- 5})}^{2} {(- 2 j^{- 3} w^{5})}^{3}}$

Step 2.2

Apply the product rule to $- 2 j^{- 3} w^{5}$ .

$\frac{1}{{(4 j^{- 5})}^{2} {(w^{- 5})}^{2} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.3

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

$\frac{1}{{(4 \frac{1}{j^{5}})}^{2} {(w^{- 5})}^{2} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.4

Combine $4$ and $\frac{1}{j^{5}}$ .

$\frac{1}{{(\frac{4}{j^{5}})}^{2} {(w^{- 5})}^{2} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.5

Apply the product rule to $\frac{4}{j^{5}}$ .

$\frac{1}{\frac{4^{2}}{{(j^{5})}^{2}} {(w^{- 5})}^{2} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.6

Raise $4$ to the power of $2$ .

$\frac{1}{\frac{16}{{(j^{5})}^{2}} {(w^{- 5})}^{2} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.7

Multiply the exponents in ${(j^{5})}^{2}$ .

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Step 2.7.1

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

$\frac{1}{\frac{16}{j^{5 \cdot 2}} {(w^{- 5})}^{2} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.7.2

Multiply $5$ by $2$ .

$\frac{1}{\frac{16}{j^{10}} {(w^{- 5})}^{2} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.8

Multiply the exponents in ${(w^{- 5})}^{2}$ .

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Step 2.8.1

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

$\frac{1}{\frac{16}{j^{10}} w^{- 5 \cdot 2} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.8.2

Multiply $- 5$ by $2$ .

$\frac{1}{\frac{16}{j^{10}} w^{- 10} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.9

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

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} {(- 2 j^{- 3})}^{3} {(w^{5})}^{3}}$

Step 2.10

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

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} {(- 2 \frac{1}{j^{3}})}^{3} {(w^{5})}^{3}}$

Step 2.11

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

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} {(\frac{- 2}{j^{3}})}^{3} {(w^{5})}^{3}}$

Step 2.12

Move the negative in front of the fraction.

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} {(- \frac{2}{j^{3}})}^{3} {(w^{5})}^{3}}$

Step 2.13

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

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Step 2.13.1

Apply the product rule to $- \frac{2}{j^{3}}$ .

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} ({(- 1)}^{3} {(\frac{2}{j^{3}})}^{3}) {(w^{5})}^{3}}$

Step 2.13.2

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

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} ({(- 1)}^{3} \frac{2^{3}}{{(j^{3})}^{3}}) {(w^{5})}^{3}}$

Step 2.14

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

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} (- \frac{2^{3}}{{(j^{3})}^{3}}) {(w^{5})}^{3}}$

Step 2.15

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

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} (- \frac{8}{{(j^{3})}^{3}}) {(w^{5})}^{3}}$

Step 2.16

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

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Step 2.16.1

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

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} (- \frac{8}{j^{3 \cdot 3}}) {(w^{5})}^{3}}$

Step 2.16.2

Multiply $3$ by $3$ .

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} (- \frac{8}{j^{9}}) {(w^{5})}^{3}}$

Step 2.17

Multiply the exponents in ${(w^{5})}^{3}$ .

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Step 2.17.1

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

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} (- \frac{8}{j^{9}}) w^{5 \cdot 3}}$

Step 2.17.2

Multiply $5$ by $3$ .

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} (- \frac{8}{j^{9}}) w^{15}}$

$\frac{1}{\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} \cdot - 1 \frac{8}{j^{9}} w^{15}}$

Step 2.18

Combine exponents.

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Step 2.18.1

Factor out negative.

$\frac{1}{- (\frac{16}{j^{10}} \cdot \frac{1}{w^{10}} \frac{8}{j^{9}} w^{15})}$

Step 2.18.2

Multiply $\frac{16}{j^{10}}$ by $\frac{1}{w^{10}}$ .

$\frac{1}{- (\frac{16}{j^{10} w^{10}} \cdot \frac{8}{j^{9}} w^{15})}$

Step 2.18.3

Multiply $\frac{16}{j^{10} w^{10}}$ by $\frac{8}{j^{9}}$ .

$\frac{1}{- (\frac{16 \cdot 8}{j^{10} w^{10} j^{9}} w^{15})}$

Step 2.18.4

Multiply $16$ by $8$ .

$\frac{1}{- (\frac{128}{j^{10} w^{10} j^{9}} w^{15})}$

Step 2.18.5

Multiply $j^{10}$ by $j^{9}$ by adding the exponents.

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Step 2.18.5.1

Move $j^{9}$ .

$\frac{1}{- (\frac{128}{j^{9} j^{10} w^{10}} w^{15})}$

Step 2.18.5.2

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

$\frac{1}{- (\frac{128}{j^{9 + 10} w^{10}} w^{15})}$

Step 2.18.5.3

Add $9$ and $10$ .

$\frac{1}{- (\frac{128}{j^{19} w^{10}} w^{15})}$

Step 2.18.6

Combine $\frac{128}{j^{19} w^{10}}$ and $w^{15}$ .

$\frac{1}{- \frac{128 w^{15}}{j^{19} w^{10}}}$

Step 2.19

Reduce the expression $\frac{128 w^{15}}{j^{19} w^{10}}$ by cancelling the common factors.

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Step 2.19.1

Factor $w^{10}$ out of $128 w^{15}$ .

$\frac{1}{- \frac{w^{10} (128 w^{5})}{j^{19} w^{10}}}$

Step 2.19.2

Factor $w^{10}$ out of $j^{19} w^{10}$ .

$\frac{1}{- \frac{w^{10} (128 w^{5})}{w^{10} j^{19}}}$

Step 2.19.3

Cancel the common factor.

$\frac{1}{- \frac{w^{10} (128 w^{5})}{w^{10} j^{19}}}$

Step 2.19.4

Rewrite the expression.

$\frac{1}{- \frac{128 w^{5}}{j^{19}}}$

Step 3

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \frac{128 w^{5}}{j^{19}}}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{1}{\frac{128 w^{5}}{j^{19}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \frac{j^{19}}{128 w^{5}})$

Step 5

Multiply $\frac{j^{19}}{128 w^{5}}$ by $1$ .

$- \frac{j^{19}}{128 w^{5}}$