Get expert insights and community support for your questions on IDNLearn.com. Our platform offers reliable and comprehensive answers to help you make informed decisions quickly and easily.

Which expression is equivalent to [tex]\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}[/tex]?

A. [tex]\sqrt[16]{4^5}[/tex]
B. [tex]\sqrt{2^5}[/tex]
C. 2
D. 4


Sagot :

To find which expression is equivalent to \(\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}\), let's break it down step-by-step.

1. Combine the exponents in the numerator:
[tex]\[ 4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}} = 4^{\left(\frac{5}{4} + \frac{1}{4}\right)} = 4^{\frac{6}{4}} = 4^{\frac{3}{2}} \][/tex]
So our expression now is:
[tex]\[ \left(\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}} \][/tex]

2. Simplify the division of the bases with exponents:
[tex]\[ \frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}} = 4^{\left(\frac{3}{2} - \frac{1}{2}\right)} = 4^{\frac{2}{2}} = 4^1 = 4 \][/tex]
Now our expression simplifies to:
[tex]\[ \left(4\right)^{\frac{1}{2}} \][/tex]

3. Take the square root of the result:
[tex]\[ \left(4\right)^{\frac{1}{2}} = \sqrt{4} = 2 \][/tex]

Thus, the equivalent expression is:
[tex]\[ 2 \][/tex]

So the correct answer is:
[tex]\[ \boxed{2} \][/tex]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.