Basic Math Examples

$\frac{125 \cdot 5^{2 n}}{5^{n + 1}} = 625$

Step 1

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

$125 \cdot 5^{2 n} \cdot 5^{- (n + 1)} = 625$

Step 2

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

$5^{3} \cdot 5^{2 n} \cdot 5^{- (n + 1)} = 625$

Step 3

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

$5^{3 + 2 n} \cdot 5^{- (n + 1)} = 625$

Step 4

Multiply $5^{3 + 2 n}$ by $5^{- (n + 1)}$ by adding the exponents.

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Step 4.1

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

$5^{3 + 2 n - (n + 1)} = 625$

Step 4.2

Simplify each term.

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Step 4.2.1

Apply the distributive property.

$5^{3 + 2 n - n - 1 \cdot 1} = 625$

Step 4.2.2

Multiply $- 1$ by $1$ .

$5^{3 + 2 n - n - 1} = 625$

Step 4.3

Subtract $1$ from $3$ .

$5^{2 n - n + 2} = 625$

Step 4.4

Subtract $n$ from $2 n$ .

$5^{n + 2} = 625$

Step 5

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

$5^{n + 2} = 5^{4}$

Step 6

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

$n + 2 = 4$

Step 7

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

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Step 7.1

Subtract $2$ from both sides of the equation.

$n = 4 - 2$

Step 7.2

Subtract $2$ from $4$ .

$n = 2$