Aubrey says that the product of [tex]$10^4$[/tex] and [tex]$10^{-2}$[/tex] is [tex]$10^{-8}$[/tex]. Is she correct? If not, explain why.
A. Yes, she is correct.
B. No, the exponent should be positive 8.
C. No, she multiplied the exponents instead of adding them.
D. No, she multiplied the exponents instead of subtracting them.
Answer (1)
Calculate the product of 1 0 4 and 1 0 − 2 by adding the exponents: 1 0 4 + ( − 2 ) .
Simplify the exponent: 4 + ( − 2 ) = 2 .
The correct product is 1 0 2 = 100 .
Aubrey is incorrect because she multiplied the exponents instead of adding them. The correct answer is 1 0 2 .
Explanation
Analyzing the Problem Aubrey claims that the product of 1 0 4 and 1 0 − 2 is 1 0 − 8 . We need to determine if she is correct and, if not, explain why.
Applying the Exponent Rule To determine if Aubrey's claim is correct, we need to calculate the product of 1 0 4 and 1 0 − 2 . Recall the rule for multiplying exponential terms with the same base: a m × a n = a m + n . In this case, we have: 1 0 4 × 1 0 − 2 = 1 0 4 + ( − 2 )
Calculating the Product Now, we simplify the exponent: 4 + ( − 2 ) = 4 − 2 = 2 So, the product is: 1 0 2 = 100 Aubrey's claim is that the product is 1 0 − 8 , which is incorrect.
Identifying the Error The correct product is 1 0 2 , which equals 100. Aubrey multiplied the exponents incorrectly. Instead of adding the exponents, she seems to have multiplied them or made a sign error. The correct operation is to add the exponents: 4 + ( − 2 ) = 2 .
Conclusion Therefore, Aubrey is incorrect. The correct answer is 1 0 2 = 100 . The reason Aubrey is wrong is that she multiplied the exponents instead of adding them.
Examples
Understanding exponent rules is crucial in many fields, such as computer science and engineering. For example, when dealing with data storage, we often encounter powers of 2 (e.g., kilobytes, megabytes, gigabytes). If a computer has 2 10 kilobytes of memory and you want to increase it by a factor of 2 2 , you would multiply these values: 2 10 × 2 2 = 2 10 + 2 = 2 12 kilobytes. This simple calculation helps determine the new memory capacity without complex computations.
