Question

If $\sqrt{2^n = 1024$ then $3^{2\left(\frac{n}{4}-1\right)} = \dots\dots\dots$}

• A. 3
• B. 9
• C. 27
• D. 81

Hint:

When solving exponential equations, try to express both sides with the same base. If the powers are equal, then their exponents must also be equal. In multiple-choice questions, if your literal calculation doesn't match any option, re-examine the question for common typos, especially in exponents or coefficients, that might lead to one of the given answers.

Answer 1

The Correct Option is D

Step 1: Solve for 'n' from the first equation.
The given equation is $\sqrt{2^n} = 1024$.
We can rewrite $\sqrt{2^n}$ as $2^{n/2}$.
So, $2^{n/2} = 1024$.
Express 1024 as a power of 2: $1024 = 2^{10}$.
Equating the exponents:
$n/2 = 10$
$n = 2 \times 10$
$n = 20$
Step 2: Substitute the value of 'n' into the expression to be evaluated.
The expression to evaluate is $3^{2\left(\frac{n}{4}-1\right)}$.
Substitute $n = 20$:
$3^{2\left(\frac{20}{4}-1\right)}$
Step 3: Simplify the exponent.
First, simplify the term inside the parenthesis: $\frac{20}{4} = 5$. So, the exponent becomes $2(5 - 1)$.
$2(5 - 1) = 2(4) = 8$.
Step 4: Calculate the final value (and address potential typo).
The expression is $3^8$.
$3^8 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 6561$.
However, $6561$ is not among the given options (3, 9, 27, 81).
Given the options, it is highly probable there is a typo in the question and the exponent should have been $\left(\frac{n}{4}-1\right)$ instead of $2\left(\frac{n}{4}-1\right)$. Let's proceed with this assumption to match one of the options.
Recalculation based on assumed typo ($3^{\left(\frac{n}{4}-1\right)}$):
Substitute $n=20$ into the modified expression:
$3^{\left(\frac{20}{4}-1\right)}$
Simplify the exponent:
$3^{(5-1)} = 3^4$
Calculate the final value:
$3^4 = 3 \times 3 \times 3 \times 3 = 81$. Step 5: Compare the result with the given options.
The calculated value $81$ matches option (4). $$(4) 81$$