which expression is equivalent to $\\left(\\frac{4^{\\frac{5}{4}}\\cdot4^{\\frac{1}{4}}}{4^{\\frac{1}{2}}}\\r…

Explanation:

Step1: Use the rule (a^m\cdot a^n=a^{m + n})

For the numerator (4^{\frac{5}{4}}\cdot4^{\frac{1}{4}}), we have (4^{\frac{5}{4}+\frac{1}{4}}=4^{\frac{5 + 1}{4}}=4^{\frac{6}{4}}=4^{\frac{3}{2}})

Step2: Use the rule (\frac{a^m}{a^n}=a^{m - n})

Now we have (\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}}=4^{\frac{3}{2}-\frac{1}{2}}=4^{\frac{3 - 1}{2}}=4^{1})

Step3: Use the rule ((a^m)^n=a^{mn})

Take (\left(4^{1}\right)^{\frac{1}{2}}), then (4^{\frac{1}{2}}=\sqrt{4} = 2)

Another way:

Step1: Express (4) as (2^2)

The original expression (\left(\frac{4^{\frac{5}{4}}\cdot4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}) becomes (\left(\frac{(2^2)^{\frac{5}{4}}\cdot(2^2)^{\frac{1}{4}}}{(2^2)^{\frac{1}{2}}}\right)^{\frac{1}{2}})

Step2: Use the rule ((a^m)^n=a^{mn})

((2^2)^{\frac{5}{4}}=2^{\frac{5}{2}}), ((2^2)^{\frac{1}{4}}=2^{\frac{1}{2}}), ((2^2)^{\frac{1}{2}}=2^{1}) The expression inside the brackets is (\frac{2^{\frac{5}{2}}\cdot2^{\frac{1}{2}}}{2^{1}})

Step3: Use the rule (a^m\cdot a^n=a^{m + n}) and (\frac{a^m}{a^n}=a^{m - n})

(\frac{2^{\frac{5}{2}+\frac{1}{2}}}{2^{1}}=\frac{2^{3}}{2^{1}}=2^{3 - 1}=2^{2})

Step4: Use the rule ((a^m)^n=a^{mn}) again

(\left(2^{2}\right)^{\frac{1}{2}}=2^{2\times\frac{1}{2}}=2)

Answer:

(2)