If $ 4 imes 4^{(n + 5)} = 256 $, then the value of $ 5^{(n + 4)}$ is:

Question

If $ 4 	imes 4^{(n + 5)} = 256 $, then the value of $ 5^{(n + 4)}$ is:

Examoogle · Accepted Answer

The correct answer is Option B: 25. Express both sides as a base of 4: 4^(1 + n + 5) = 4^4. Solve for exponent: 6 + n = 4, so n = -2. Then calculate 5^(n+4) = 5^(-2+4) = 5^2 = 25.

Step-by-Step Solution & Concepts:
Shortcut Trick: Express both sides as a base of 4: 4^(1 + n + 5) = 4^4. Solve for exponent: 6 + n = 4, so n = -2. Then calculate 5^(n+4) = 5^(-2+4) = 5^2 = 25.

Formula Used: $$a^m 	imes a^n = a^{m+n}$$

Given: 4 * 4^(n+5) = 256

Steps: Simplify the given equation: Using the law of exponents $a^m 	imes a^n = a^{m+n}$, rewrite the left side: $4^1 	imes 4^{n+5} = 4^{n+6}$.. Equate the bases: Since $256 = 4^4$, we can set the exponents equal to each other.. Solve for $n$: Equating the powers: $n + 6 = 4$, which results in $n = -2$.. Find the final value: Substitute $n = -2$ into $5^{n+4}$: $5^{-2+4} = 5^2 = 25$.

Answer: B is correct because solving the exponential equation $4^{n+6} = 256$ gives $n = -2$, and calculating $5^{-2+4}$ yields $25$.

Exam Tip: This concept of equating bases is foundational for solving logarithmic and more complex algebraic equations in competitive exams.

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