Trigonometry Examples

$\frac{12 r^{- 5} {(r^{2})}^{5}}{19 {(r^{- 4})}^{2}}$

Step 1

Simplify the expression.

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Step 1.1

Move $r^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12 {(r^{2})}^{5}}{19 {(r^{- 4})}^{2} r^{5}}$

Step 1.2

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

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Step 1.2.1

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

$\frac{12 r^{2 \cdot 5}}{19 {(r^{- 4})}^{2} r^{5}}$

Step 1.2.2

Multiply $2$ by $5$ .

$\frac{12 r^{10}}{19 {(r^{- 4})}^{2} r^{5}}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

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

$\frac{12 r^{10}}{19 r^{- 4 \cdot 2} r^{5}}$

Step 2.1.2

Multiply $- 4$ by $2$ .

$\frac{12 r^{10}}{19 r^{- 8} r^{5}}$

Step 2.2

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

$\frac{12 r^{10}}{19 \frac{1}{r^{8}} r^{5}}$

Step 2.3

Combine exponents.

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Step 2.3.1

Combine $19$ and $\frac{1}{r^{8}}$ .

$\frac{12 r^{10}}{\frac{19}{r^{8}} r^{5}}$

Step 2.3.2

Combine $\frac{19}{r^{8}}$ and $r^{5}$ .

$\frac{12 r^{10}}{\frac{19 r^{5}}{r^{8}}}$

Step 2.4

Reduce the expression $\frac{19 r^{5}}{r^{8}}$ by cancelling the common factors.

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Step 2.4.1

Factor $r^{5}$ out of $19 r^{5}$ .

$\frac{12 r^{10}}{\frac{r^{5} \cdot 19}{r^{8}}}$

Step 2.4.2

Factor $r^{5}$ out of $r^{8}$ .

$\frac{12 r^{10}}{\frac{r^{5} \cdot 19}{r^{5} r^{3}}}$

Step 2.4.3

Cancel the common factor.

$\frac{12 r^{10}}{\frac{r^{5} \cdot 19}{r^{5} r^{3}}}$

Step 2.4.4

Rewrite the expression.

$\frac{12 r^{10}}{\frac{19}{r^{3}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$12 r^{10} \frac{r^{3}}{19}$

Step 4

Multiply $12 r^{10} \frac{r^{3}}{19}$ .

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Step 4.1

Combine $12$ and $\frac{r^{3}}{19}$ .

$r^{10} \frac{12 r^{3}}{19}$

Step 4.2

Combine $r^{10}$ and $\frac{12 r^{3}}{19}$ .

$\frac{r^{10} (12 r^{3})}{19}$

Step 4.3

Multiply $r^{10}$ by $r^{3}$ by adding the exponents.

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Step 4.3.1

Move $r^{3}$ .

$\frac{r^{3} r^{10} \cdot 12}{19}$

Step 4.3.2

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

$\frac{r^{3 + 10} \cdot 12}{19}$

Step 4.3.3

Add $3$ and $10$ .

$\frac{r^{13} \cdot 12}{19}$

Step 5

Move $12$ to the left of $r^{13}$ .

$\frac{12 r^{13}}{19}$