Which of the following is equivalent to (16^(3/2))^(1/2)?
The correct answer and explanation is:
Correct Answer: (163/2)1/2=163/4(16^{3/2})^{1/2} = 16^{3/4}
Step-by-Step Explanation:
We are given the expression: (163/2)1/2(16^{3/2})^{1/2}
This is a power of a power situation. In such expressions, you multiply the exponents: (163/2)1/2=16(3/2)⋅(1/2)=163/4(16^{3/2})^{1/2} = 16^{(3/2) \cdot (1/2)} = 16^{3/4}
So, the expression simplifies to: 163/4\boxed{16^{3/4}}
Understanding the Expression:
To understand what this means numerically:
- Start with the base, which is 16.
- The exponent 3/4 means we:
- First take the fourth root of 16 (because of the denominator 4),
- Then raise the result to the third power (because of the numerator 3).
Step-by-step:
- 164=2\sqrt[4]{16} = 2, because 24=162^4 = 16
- 23=82^3 = 8
Therefore, the expression evaluates to: (163/2)1/2=163/4=8(16^{3/2})^{1/2} = 16^{3/4} = 8
Why This Is Important in Algebra:
Understanding how to manipulate exponents is essential in algebra, especially when dealing with radicals and exponential functions. The key rules used here include:
- Power of a Power Rule: (am)n=amn(a^m)^n = a^{mn}
- Roots and Fractional Exponents: a1/n=ana^{1/n} = \sqrt[n]{a}
These rules allow you to simplify complex expressions and solve exponential equations more easily. In science, engineering, and real-world problem-solving, knowing how to handle exponents quickly and correctly is very useful.
Thus, the expression (163/2)1/2(16^{3/2})^{1/2} is equivalent to 163/4\boxed{16^{3/4}}.