Question

Simplify the following expression: \[ \frac{2^{n+5} - 4 \cdot 2^{n}}{2 \cdot (2^{n+4})}. \]

• A. \( 2^{n+1} - \frac{1}{4} \)
• B. \( 2^{n+1} \)
• C. \( -2^{n+1} \)
• D. \( \frac{7}{8} \)

Hint:

When simplifying powers of 2, factor common terms and reduce the expression.

Answer 1

The Correct Option is D

Step 1: The given expression is: \[ \frac{2^{n+5} - 4 \cdot 2^{n}}{2 \cdot (2^{n+4})}. \] Step 2: Simplify the numerator \( 2^{n+5} - 4 \cdot 2^{n} \): \[ 2^{n+5} = 2^{n} \cdot 2^5 = 32 \cdot 2^{n}, \quad 4 \cdot 2^{n} = 2^2 \cdot 2^{n} = 2^{n+2}. \] Thus, the numerator becomes: \[ 32 \cdot 2^{n} - 2^{n+2} = 2^{n} (32 - 4) = 28 \cdot 2^{n}. \] Step 3: Simplify the denominator \( 2 \cdot 2^{n+4} \): \[ 2 \cdot 2^{n+4} = 2 \cdot 2^{n} \cdot 2^4 = 2 \cdot 2^{n} \cdot 16 = 16 \cdot 2^{n}. \] Step 4: Now substitute the simplified numerator and denominator into the original expression: \[ \frac{28 \cdot 2^{n}}{16 \cdot 2^{n}} = \frac{28}{16} = \frac{7}{8}. \]