Question

Quantity A: \( \frac{2^{30} - 2^{29}}{2} \)
Quantity B: \( 2^{28} \)

• A. Quantity A is greater.
• B. Quantity B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.
• E.

Hint:

When you see a sum or difference of terms with exponents (like \( a^x \pm a^y \)), your first instinct should be to factor out the term with the smaller exponent. This is a very common technique for simplifying such expressions.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks us to compare two quantities involving exponents. To do this, we need to simplify Quantity A using the rules of exponents.
Step 2: Key Formula or Approach:
The key exponent rules needed here are:
1. Factoring: \( a^m - a^n \) can often be simplified by factoring out the smaller power.
2. The law of exponents for division: \( \frac{a^m}{a^n} = a^{m-n} \).
3. The law of exponents for multiplication: \( a^m \times a^n = a^{m+n} \).
Step 3: Detailed Explanation:
Let's focus on simplifying the expression for Quantity A.
\[ \text{Quantity A} = \frac{2^{30} - 2^{29}}{2} \] First, simplify the numerator, \( 2^{30} - 2^{29} \). We can factor out the term with the smaller exponent, which is \( 2^{29} \).
Note that \( 2^{30} \) can be rewritten as \( 2^{1} \times 2^{29} \).
\[ 2^{30} - 2^{29} = (2^1 \times 2^{29}) - (1 \times 2^{29}) \] Now, factor out \( 2^{29} \):
\[ = 2^{29} \times (2 - 1) \] \[ = 2^{29} \times 1 = 2^{29} \] So, the numerator simplifies to \( 2^{29} \). Now, substitute this back into the expression for Quantity A:
\[ \text{Quantity A} = \frac{2^{29}}{2} \] Since \( 2 \) is the same as \( 2^1 \), we can use the division rule for exponents:
\[ \text{Quantity A} = \frac{2^{29}}{2^1} = 2^{29-1} = 2^{28} \] Now we compare the simplified Quantity A with Quantity B.
- Quantity A = \( 2^{28} \)
- Quantity B = \( 2^{28} \)
The two quantities are identical.
Step 4: Final Answer:
After simplification, Quantity A is equal to \( 2^{28} \), which is the same as Quantity B. Therefore, the two quantities are equal. The correct choice is (C).