Question

The value of $x$ in the expression \[ \left(\frac{2^{2x+1} \cdot 4^{x-2}}{8^{x-3}}\right) = 1024 \] is:

• A. 4
• B. 5
• C. 6
• D. 7

Hint:

Convert all bases to the same number before solving exponent equations.

Answer 1

The Correct Option is A

Concept: Convert all terms to base 2.
Step 1: Rewrite powers.
\[ 4 = 2^2,\quad 8 = 2^3,\quad 1024 = 2^{10} \]
Step 2: Substitute.
\[ \frac{2^{2x+1} \cdot (2^2)^{x-2}}{(2^3)^{x-3}} = 2^{10} \]
Step 3: Simplify exponents.
\[ = \frac{2^{2x+1} \cdot 2^{2x-4}}{2^{3x-9}} = 2^{10} \] \[ = 2^{(2x+1 + 2x-4 - (3x-9))} \] \[ = 2^{(4x -3 -3x +9)} = 2^{(x +6)} \]
Step 4: Equate powers.
\[ x + 6 = 10 \Rightarrow x = 4 \]
Hence, the value of $x$ is 4.