Pre-Algebra Examples

$\frac{5^{2 n + 1} - 25^{n}}{5^{2 n + 2}}$

Step 1

Simplify and reorder the polynomial.

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Step 1.1

Split the fraction $\frac{5^{2 n + 1} - 25^{n}}{5^{2 n + 2}}$ into two fractions.

$\frac{5^{2 n + 1}}{5^{2 n + 2}} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2

Simplify each term.

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Step 1.2.1

Cancel the common factor of $5^{2 n + 1}$ and $5^{2 n + 2}$ .

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Step 1.2.1.1

Factor $5^{2 n + 2}$ out of $5^{2 n + 1}$ .

$\frac{5^{2 n + 2} \cdot 5^{2 n + 1 - (2 n + 2)}}{5^{2 n + 2}} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.1.2

Cancel the common factors.

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Step 1.2.1.2.1

Multiply by $1$ .

$\frac{5^{2 n + 2} \cdot 5^{2 n + 1 - (2 n + 2)}}{5^{2 n + 2} \cdot 1} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.1.2.2

Cancel the common factor.

$\frac{5^{2 n + 2} \cdot 5^{2 n + 1 - (2 n + 2)}}{5^{2 n + 2} \cdot 1} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.1.2.3

Rewrite the expression.

$\frac{5^{2 n + 1 - (2 n + 2)}}{1} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.1.2.4

Divide $5^{2 n + 1 - (2 n + 2)}$ by $1$ .

$5^{2 n + 1 - (2 n + 2)} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Apply the distributive property.

$5^{2 n + 1 - (2 n) - 1 \cdot 2} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.2.2

Multiply $2$ by $- 1$ .

$5^{2 n + 1 - 2 n - 1 \cdot 2} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.2.3

Multiply $- 1$ by $2$ .

$5^{2 n + 1 - 2 n - 2} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.3

Combine the opposite terms in $2 n + 1 - 2 n - 2$ .

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Step 1.2.3.1

Subtract $2 n$ from $2 n$ .

$5^{0 + 1 - 2} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.3.2

Add $0$ and $1$ .

$5^{1 - 2} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.4

Subtract $2$ from $1$ .

$5^{- 1} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.5

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

$\frac{1}{5} + \frac{- 25^{n}}{5^{2 n + 2}}$

Step 1.2.6

Move the negative in front of the fraction.

$\frac{1}{5} - \frac{25^{n}}{5^{2 n + 2}}$

Step 2

The degree cannot be determined because $\frac{1}{5} - \frac{25^{n}}{5^{2 n + 2}}$ is not a polynomial.

Not a polynomial