Precalculus Examples

$625^{- 3 r} \cdot 125 = {(\frac{1}{125})}^{- r}$

Step 1

Apply the product rule to $\frac{1}{125}$ .

$625^{- 3 r} \cdot 125 = \frac{1^{- r}}{125^{- r}}$

Step 2

One to any power is one.

$625^{- 3 r} \cdot 125 = \frac{1}{125^{- r}}$

Step 3

Move $125^{- r}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$625^{- 3 r} \cdot 125 = 125^{r}$

Step 4

Rewrite $625$ as $5^{4}$ .

${(5^{4})}^{- 3 r} \cdot 125 = 125^{r}$

Step 5

Multiply the exponents in ${(5^{4})}^{- 3 r}$ .

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Step 5.1

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

$5^{4 (- 3 r)} \cdot 125 = 125^{r}$

Step 5.2

Multiply $- 3$ by $4$ .

$5^{- 12 r} \cdot 125 = 125^{r}$

Step 6

Rewrite $125$ as $5^{3}$ .

$5^{- 12 r} \cdot 5^{3} = 125^{r}$

Step 7

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

$5^{- 12 r + 3} = 125^{r}$

Step 8

Create equivalent expressions in the equation that all have equal bases.

$5^{- 12 r + 3} = 5^{3 r}$

Step 9

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$- 12 r + 3 = 3 r$

Step 10

Solve for $r$ .

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Step 10.1

Move all terms containing $r$ to the left side of the equation.

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Step 10.1.1

Subtract $3 r$ from both sides of the equation.

$- 12 r + 3 - 3 r = 0$

Step 10.1.2

Subtract $3 r$ from $- 12 r$ .

$- 15 r + 3 = 0$

Step 10.2

Subtract $3$ from both sides of the equation.

$- 15 r = - 3$

Step 10.3

Divide each term in $- 15 r = - 3$ by $- 15$ and simplify.

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Step 10.3.1

Divide each term in $- 15 r = - 3$ by $- 15$ .

$\frac{- 15 r}{- 15} = \frac{- 3}{- 15}$

Step 10.3.2

Simplify the left side.

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Step 10.3.2.1

Cancel the common factor of $- 15$ .

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Step 10.3.2.1.1

Cancel the common factor.

$\frac{- 15 r}{- 15} = \frac{- 3}{- 15}$

Step 10.3.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 3}{- 15}$

Step 10.3.3

Simplify the right side.

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Step 10.3.3.1

Cancel the common factor of $- 3$ and $- 15$ .

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Step 10.3.3.1.1

Factor $- 3$ out of $- 3$ .

$r = \frac{- 3 (1)}{- 15}$

Step 10.3.3.1.2

Cancel the common factors.

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Step 10.3.3.1.2.1

Factor $- 3$ out of $- 15$ .

$r = \frac{- 3 \cdot 1}{- 3 \cdot 5}$

Step 10.3.3.1.2.2

Cancel the common factor.

$r = \frac{- 3 \cdot 1}{- 3 \cdot 5}$

Step 10.3.3.1.2.3

Rewrite the expression.

$r = \frac{1}{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$r = \frac{1}{5}$

Decimal Form:

$r = 0.2$