Precalculus Examples

$\frac{625^{- 3 n}}{125^{2 - 3 n}} = 125$

Step 1

Move $125^{2 - 3 n}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$625^{- 3 n} \cdot 125^{- (2 - 3 n)} = 125$

Step 2

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

${(5^{4})}^{- 3 n} \cdot 125^{- (2 - 3 n)} = 125$

Step 3

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

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Step 3.1

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

$5^{4 (- 3 n)} \cdot 125^{- (2 - 3 n)} = 125$

Step 3.2

Multiply $- 3$ by $4$ .

$5^{- 12 n} \cdot 125^{- (2 - 3 n)} = 125$

Step 4

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

$5^{- 12 n} \cdot {(5^{3})}^{- (2 - 3 n)} = 125$

Step 5

Multiply the exponents in ${(5^{3})}^{- (2 - 3 n)}$ .

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Step 5.1

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

$5^{- 12 n} \cdot 5^{3 (- (2 - 3 n))} = 125$

Step 5.2

Apply the distributive property.

$5^{- 12 n} \cdot 5^{3 (- 1 \cdot 2 - (- 3 n))} = 125$

Step 5.3

Multiply $- 1$ by $2$ .

$5^{- 12 n} \cdot 5^{3 (- 2 - (- 3 n))} = 125$

Step 5.4

Multiply $- 3$ by $- 1$ .

$5^{- 12 n} \cdot 5^{3 (- 2 + 3 n)} = 125$

Step 5.5

Apply the distributive property.

$5^{- 12 n} \cdot 5^{3 \cdot - 2 + 3 (3 n)} = 125$

Step 5.6

Multiply $3$ by $- 2$ .

$5^{- 12 n} \cdot 5^{- 6 + 3 (3 n)} = 125$

Step 5.7

Multiply $3$ by $3$ .

$5^{- 12 n} \cdot 5^{- 6 + 9 n} = 125$

Step 6

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

$5^{- 12 n - 6 + 9 n} = 125$

Step 7

Add $- 12 n$ and $9 n$ .

$5^{- 3 n - 6} = 125$

Step 8

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

$5^{- 3 n - 6} = 5^{3}$

Step 9

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

$- 3 n - 6 = 3$

Step 10

Solve for $n$ .

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Step 10.1

Move all terms not containing $n$ to the right side of the equation.

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Step 10.1.1

Add $6$ to both sides of the equation.

$- 3 n = 3 + 6$

Step 10.1.2

Add $3$ and $6$ .

$- 3 n = 9$

Step 10.2

Divide each term in $- 3 n = 9$ by $- 3$ and simplify.

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Step 10.2.1

Divide each term in $- 3 n = 9$ by $- 3$ .

$\frac{- 3 n}{- 3} = \frac{9}{- 3}$

Step 10.2.2

Simplify the left side.

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Step 10.2.2.1

Cancel the common factor of $- 3$ .

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Step 10.2.2.1.1

Cancel the common factor.

$\frac{- 3 n}{- 3} = \frac{9}{- 3}$

Step 10.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{9}{- 3}$

Step 10.2.3

Simplify the right side.

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Step 10.2.3.1

Divide $9$ by $- 3$ .

$n = - 3$