Consider the algebraic fraction below: \frac{5^{n+2} - 5^n}{5^{n+1} - 5^n} Which of the fo… — State PSC Exam

Consider the algebraic fraction below:

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

Which of the following represents the simplified exact value of this expression?

The expression can be systematically simplified by factoring out the lowest common power of the base, which is

5^n

, from both the numerator and the denominator. The numerator becomes

5^n(5^2 - 1) = 5^n(24)

. The denominator becomes

5^n(5^1 - 1) = 5^n(4)

. Canceling the common factor

5^n

from the top and bottom leaves

24 / 4

, which equals 6.

Identify the greatest common factor in both the numerator and the denominator.

The common factor is

5^n

Factoring out the lowest power of the base simplifies algebraic expressions involving exponential terms.

Factor out

5^n

from the numerator.

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

Applying the exponent rule

a^{x+y} = a^x \times a^y

in reverse allows us to extract the common term.

Factor out

5^n

from the denominator.

5^{n+1} - 5^n = 5^n(5^1 - 1)

Applying the same extraction process to the bottom expression.

Substitute the factored forms back into the fraction and cancel the common

5^n

term.

\frac{5^n(25 - 1)}{5^n(5 - 1)} = \frac{24}{4}

The

5^n

multiplier in the numerator and denominator cancel each other out, leaving basic arithmetic values.

Perform the final arithmetic division.

24 \div 4 = 6

Simplifying the remaining numerical fraction yields the final exact value.

Simplifying exponential expressions by factoring out common base powers.

Estimated Time:1m 15s

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