Simplify the expression.
$\frac{b^{-\frac{1}{2}} b^{\frac{5}{2}}}{b^{\frac{1}{3}}}$
Write your answer using only positive exponents.
Assume that all variables are positive real numbers.
Answer (2)
Simplify the numerator using the property b m b n = b m + n : b − 2 1 b 2 5 = b 2 .
Simplify the expression using the property b n b m = b m − n : b 3 1 b 2 = b 2 − 3 1 .
Simplify the exponent: 2 − 3 1 = 3 5 .
The simplified expression is b 3 5 .
Explanation
Understanding the Problem We are asked to simplify the expression b 3 1 b − 2 1 b 2 5 and write the answer using only positive exponents. We are also told that b is a positive real number.
Simplifying the Numerator First, we simplify the numerator using the property b m b n = b m + n . In our case, we have b − 2 1 b 2 5 = b − 2 1 + 2 5 = b 2 5 − 2 1 = b 2 4 = b 2 .
Simplifying the Expression Now, we have b 3 1 b 2 . We use the property b n b m = b m − n to simplify the expression further. So, we have b 3 1 b 2 = b 2 − 3 1 .
Simplifying the Exponent Next, we simplify the exponent: 2 − 3 1 = 3 6 − 3 1 = 3 5 . Therefore, the simplified expression is b 3 5 .
Final Answer The simplified expression is b 3 5 . Since the exponent is positive, we have our final answer.
Examples
Understanding how to simplify expressions with exponents is crucial in various fields, such as physics and engineering. For example, when dealing with wave functions in quantum mechanics, you often encounter complex expressions involving exponents. Simplifying these expressions allows for easier manipulation and interpretation of the wave function, leading to a better understanding of the behavior of particles at the quantum level. Similarly, in electrical engineering, simplifying expressions with exponents can help in analyzing the behavior of circuits and signals.
To simplify the expression b 3 1 b − 2 1 b 2 5 , we first simplify the numerator to get b 2 . Then, we apply the quotient rule to obtain b 2 − 3 1 , which simplifies to b 3 5 . The final answer is b 3 5 .
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