Question

\((2^{10} - 2^9)(2^8 - 2^7) =\)

• A. 2
• B. \(2^2\)
• C. \(2^4\)
• D. \(2^8\)
• E. \(2^{16}\)

Hint:

When subtracting exponential terms with the same base, always factor out the term with the smaller exponent. This simplifies the expression dramatically and avoids large, error-prone calculations.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This problem involves simplifying an expression with exponents. The key is to factor out the common terms within each set of parentheses before multiplying.
Step 2: Key Formula or Approach:
1. Factoring: \(a^m - a^n = a^n(a^{m-n} - 1)\) for \(m>n\).
2. Multiplication of powers: \(a^m \times a^n = a^{m+n}\).
Step 3: Detailed Explanation:
1. Simplify the first term: \((2^{10} - 2^9)\)
Factor out the lowest power of 2, which is \(2^9\).
\[ 2^{10} - 2^9 = 2^9(2^{10-9} - 1) = 2^9(2^1 - 1) = 2^9(2 - 1) = 2^9(1) = 2^9 \]
2. Simplify the second term: \((2^8 - 2^7)\)
Factor out the lowest power of 2, which is \(2^7\).
\[ 2^8 - 2^7 = 2^7(2^{8-7} - 1) = 2^7(2^1 - 1) = 2^7(2 - 1) = 2^7(1) = 2^7 \]
3. Multiply the simplified terms.
The original expression becomes: \( (2^9) \times (2^7) \).
Using the rule for multiplying powers with the same base, we add the exponents:
\[ 2^9 \times 2^7 = 2^{9+7} = 2^{16} \]
Step 4: Final Answer:
The value of the expression is \(2^{16}\).