Basic Math Examples

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

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

Step 1

Move

5^{n + 1}

$5^{n + 1}$ to the numerator using the negative exponent rule

\frac{1}{b^{- n}} = b^{n}

$\frac{1}{b^{- n}} = b^{n}$ .

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

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

Step 2

Rewrite

125

$125$ as

5^{3}

$5^{3}$ .

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

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

Step 3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 4

Multiply

5^{3 + 2 n}

$5^{3 + 2 n}$ by

5^{- (n + 1)}

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

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Step 4.1

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 4.2.2

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

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

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

Step 4.3

Subtract

1

$1$ from

3

$3$ .

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

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

Step 4.4

Subtract

n

$n$ from

2 n

$2 n$ .

5^{n + 2} = 625

$5^{n + 2} = 625$

5^{n + 2} = 625

$5^{n + 2} = 625$

Step 5

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

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

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

Step 6

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

n + 2 = 4

$n + 2 = 4$

Step 7

Move all terms not containing

n

$n$ to the right side of the equation.

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Step 7.1

Subtract

2

$2$ from both sides of the equation.

n = 4 - 2

$n = 4 - 2$

Step 7.2

Subtract

2

$2$ from

4

$4$ .

n = 2

$n = 2$

n = 2

$n = 2$