Question

Quantity A: \(2^{60}\)
Quantity B: \(8^{20}\)

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

Hint:

When comparing expressions with exponents, always look for an opportunity to make the bases equal. Recognizing common powers (like \(8 = 2^3\), \(9 = 3^2\), \(27 = 3^3\), etc.) is a fundamental skill for solving such problems quickly.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks for a comparison between two numbers expressed in exponential form. To compare them, it's best to rewrite them so that they share a common base.
Step 2: Key Formula or Approach:
We will use the exponent rule known as the "power of a power" rule, which states that \((a^m)^n = a^{m \times n}\).
Step 3: Detailed Explanation:
Quantity A is given as \(2^{60}\).
Quantity B is given as \(8^{20}\).
We can express the base of Quantity B, which is 8, as a power of 2: \[ 8 = 2^3 \] Now, we substitute this back into the expression for Quantity B: \[ 8^{20} = (2^3)^{20} \] Using the power of a power rule, we multiply the exponents: \[ (2^3)^{20} = 2^{3 \times 20} = 2^{60} \] So, Quantity B is equivalent to \(2^{60}\).
Step 4: Final Answer:
By simplifying Quantity B, we have:
Quantity A = \(2^{60}\)
Quantity B = \(2^{60}\)
Both quantities are identical. Therefore, the two quantities are equal.