Question

which expression is equivalent to \\(\left(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\
ight)^2\\)?
\\(\circ \sqrt4{2^3}\\)
\\(\circ \sqrt{2^5}\\)
\\(\circ \sqrt4{4^3}\\)
\\(\circ \sqrt{4^5}\\)

Explanation:

Step1: Simplify the exponents inside

First, simplify the exponents of the base 2. We have \(2^{\frac{1}{2}} \cdot 2^{\frac{3}{4}}\). Using the rule \(a^m \cdot a^n = a^{m + n}\), so \(\frac{1}{2}+\frac{3}{4}=\frac{2 + 3}{4}=\frac{5}{4}\). Then we raise this to the power of 2: \((2^{\frac{5}{4}})^2\).

Step2: Apply the power - of - a - power rule

Using the rule \((a^m)^n=a^{m\times n}\), we get \(2^{\frac{5}{4}\times2}=2^{\frac{5}{2}}\).

Step3: Rewrite the exponent in radical form

Recall that \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\). So \(2^{\frac{5}{2}}=\sqrt{2^5}\) (since \(n = 2\) and \(m = 5\)).

Answer:

\(\sqrt{2^{5}}\) (the second option from the left)