b.) $3^{-\frac{1}{2}} \times \sqrt{3}$
Answer (2)
Rewrite 3 as 3 2 1 .
The expression becomes 3 − 2 1 × 3 2 1 .
Use the property a m × a n = a m + n to simplify the expression to 3 − 2 1 + 2 1 = 3 0 .
Simplify further to get the final answer: 1 .
Explanation
Understanding the problem We are given the expression 3 − 2 1 × 3 and our goal is to simplify it.
Rewriting the expression First, let's rewrite the square root of 3 using exponents. Recall that x = x 2 1 . Therefore, we can rewrite the expression as: 3 − 2 1 × 3 2 1
Applying exponent rules Now, we can use the property of exponents that states a m × a n = a m + n . Applying this property, we get: 3 − 2 1 + 2 1 = 3 0
Simplifying to the final answer Finally, recall that any non-zero number raised to the power of 0 is equal to 1. Therefore, 3 0 = 1 .
Examples
Understanding exponential expressions like this is crucial in many fields, such as calculating growth rates or decay in biology, finance, and physics. For instance, if a population of bacteria doubles every hour, we can use exponential expressions to predict its size after a certain number of hours. Similarly, in finance, compound interest calculations rely heavily on exponential functions to determine the future value of an investment. This problem demonstrates a fundamental property of exponents that simplifies complex calculations in these real-world scenarios.
To simplify 3 − 2 1 × 3 , we rewrite 3 as 3 2 1 and then apply the exponent rule to combine the terms. This results in 3 0 , which equals 1.
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