which expression is equivalent to $left(2^{\frac{1}{2}}cdot2^{\frac{3}{4}}\right)^{2}$?\n$sqrt4{2^{3}}$\n$sqr…

Answer:

B. $\sqrt{2^{5}}$

Explanation:

Step1: Use exponent - product rule

When multiplying numbers with the same base (a^m\cdot a^n=a^{m + n}), so (2^{\frac{1}{2}}\cdot2^{\frac{3}{4}}=2^{\frac{1}{2}+\frac{3}{4}}). [2^{\frac{1}{2}+\frac{3}{4}}=2^{\frac{2 + 3}{4}}=2^{\frac{5}{4}}]

Step2: Use power - of - a - power rule

((a^m)^n=a^{mn}), so ((2^{\frac{5}{4}})^2=2^{\frac{5}{4}\times2}). [2^{\frac{5}{4}\times2}=2^{\frac{5}{2}}]

Step3: Rewrite in radical form

The fractional - exponent rule (a^{\frac{m}{n}}=\sqrt[n]{a^{m}}), for (a = 2), (m = 5), (n=2), (2^{\frac{5}{2}}=\sqrt{2^{5}}).