Evaluate $3^{-2}$ to the nearest hundredth.
Answer (1)
Rewrite the expression using the property of negative exponents: 3 − 2 = 3 2 1 .
Calculate 3 2 = 9 , so the expression becomes 9 1 .
Convert 9 1 to a decimal, which is approximately 0.111111...
Round the decimal to the nearest hundredth: 0.11 .
Explanation
Understanding the Problem The problem asks us to evaluate 3 − 2 to the nearest hundredth. This means we need to calculate the value of 3 − 2 and then round the result to two decimal places.
Rewriting with Positive Exponent First, let's rewrite 3 − 2 using the property of negative exponents: a − n = a n 1 . So, we have: 3 − 2 = 3 2 1
Calculating the Square Next, we calculate 3 2 , which is 3 × 3 = 9 . Therefore, we have: 3 − 2 = 9 1
Converting to Decimal Now, we need to convert 9 1 to a decimal. Dividing 1 by 9 gives us approximately 0.111111...
Rounding to Nearest Hundredth Finally, we round 0.111111... to the nearest hundredth. The hundredth place is the second digit after the decimal point. Since the digit in the thousandth place is 1 (which is less than 5), we round down, keeping the hundredth place as 1. Thus, the result is 0.11.
Final Answer Therefore, 3 − 2 evaluated to the nearest hundredth is 0.11 .
Examples
Understanding exponents and their decimal representations is useful in many areas, such as calculating compound interest. For example, if you invest $100 at an annual interest rate of 3% compounded quarterly, the amount after one year is 100 ( 1 + 0.03/4 ) 4 . The term ( 1 + 0.03/4 ) 4 involves exponents and can be approximated using decimals to find the final amount.
