Select the correct answer.
Which number is the same as $\left(4^{-1}\right)\left(2^3 \div 4^{-2}\right) ?$
A. -2
B. $\frac{1}{8}$
C. 2
D. 32
E. 512
Answer (1)
Rewrite the expression using exponent rules.
Express all terms as powers of 2: 4 − 1 = 2 − 2 and 4 − 2 = 2 − 4 .
Simplify the division: 2 3 o b re ak ÷ 2 − 4 = 2 7 .
Simplify the multiplication: 2 − 2 o b re ak × 2 7 = 2 5 = 32 .
The correct answer is 32 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 4 − 1 ) l e f t ( 2 3 ÷ 4 − 2 r i g h t ) and choose the correct answer from the given options.
Expressing as Powers of 2 First, let's rewrite the expression using exponent rules. We know that 4 = 2 2 , so we can express all terms as powers of 2: 4 − 1 = ( 2 2 ) − 1 = 2 − 2 and 4 − 2 = ( 2 2 ) − 2 = 2 − 4 .
Substitution Now, substitute these into the expression: ( 2 − 2 ) ( 2 3 ÷ 2 − 4 ) .
Division of Exponents Using the rule for division of exponents, a m ÷ a n = a m − n , we have: 2 3 ÷ 2 − 4 = 2 3 − ( − 4 ) = 2 3 + 4 = 2 7 .
Substitution Substitute this back into the expression: ( 2 − 2 ) ( 2 7 ) .
Multiplication of Exponents Using the rule for multiplication of exponents, a m × a n = a m + n , we have: 2 − 2 × 2 7 = 2 − 2 + 7 = 2 5 .
Calculation Now, calculate 2 5 : 2 5 = 2 × 2 × 2 × 2 × 2 = 32 .
Final Answer Therefore, the simplified expression is equal to 32. Comparing this to the options, we see that the correct answer is D.
Examples
Understanding and simplifying exponential expressions is crucial in various fields, such as computer science, where data storage and processing speeds are often expressed in powers of 2. For instance, a computer's memory capacity might be 2 32 bytes, or 4 gigabytes. Similarly, in physics, exponential relationships are used to describe radioactive decay, where the amount of a substance decreases exponentially over time. Knowing how to manipulate exponents allows scientists and engineers to make accurate predictions and calculations in these scenarios.
