Which expression is equivalent to
The Correct Answer and Explanation is:
Let’s evaluate the expression step-by-step.
We are given: (454⋅414412)12\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}(421445⋅441)21
Step 1: Simplify the numerator using laws of exponents
When multiplying expressions with the same base, add the exponents: 454⋅414=454+14=464=4324^{\frac{5}{4}} \cdot 4^{\frac{1}{4}} = 4^{\frac{5}{4} + \frac{1}{4}} = 4^{\frac{6}{4}} = 4^{\frac{3}{2}}445⋅441=445+41=446=423
So the whole expression becomes: (432412)12\left(\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}(421423)21
Step 2: Simplify the fraction
When dividing powers with the same base, subtract the exponents: 432412=432−12=41\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}} = 4^{\frac{3}{2} – \frac{1}{2}} = 4^{1}421423=423−21=41
Now the expression is: (41)12=412=4=2(4^1)^{\frac{1}{2}} = 4^{\frac{1}{2}} = \sqrt{4} = 2(41)21=421=4=2
Final Answer:
2\boxed{2}2
Explanation
This problem tests your understanding of the laws of exponents and how to simplify radical and exponential expressions. You’re given a complex expression involving exponents and a radical, and the goal is to simplify it step-by-step using exponent rules.
We begin with the expression inside the parentheses: 454⋅414412\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}421445⋅441
Using the rule for multiplying powers with the same base, we add the exponents: 454+14=464=4324^{\frac{5}{4} + \frac{1}{4}} = 4^{\frac{6}{4}} = 4^{\frac{3}{2}}445+41=446=423
Next, we divide by 4124^{\frac{1}{2}}421. Again, applying the law for dividing powers: 432÷412=432−12=414^{\frac{3}{2}} \div 4^{\frac{1}{2}} = 4^{\frac{3}{2} – \frac{1}{2}} = 4^1423÷421=423−21=41
Now the entire expression becomes (41)12(4^1)^{\frac{1}{2}}(41)21. When you raise a power to a power, you multiply the exponents: (41)12=412=4=2(4^1)^{\frac{1}{2}} = 4^{\frac{1}{2}} = \sqrt{4} = 2(41)21=421=4=2
Thus, the simplified expression is equal to 2. This shows the power of understanding exponent rules to break down and simplify complex-looking expressions into basic numerical answers.
Correct answer:
