Answer
C - the fourth root of 2
Explanation
First, we are going to write our expression in mathematical notation:
2 to the 3 over 4 power, all over 2 to the 1 over 2 power = [tex]\frac{2^{\frac{3}{4} }}{2^{\frac{1}{2} }}[/tex]
Now, we are going to use the law of exponents for division: [tex]\frac{a^m}{a^n} =a^{m-n}[/tex]
We can infer from our expression that [tex]a=2[/tex], [tex]m=\frac{3}{4}[/tex], and [tex]n=\frac{1}{2}[/tex], so let's use our rule:
[tex]\frac{2^{\frac{3}{4} }}{2^{\frac{1}{2} }}=2^{\frac{3}{4}-\frac{1}{2} }}=2^{\frac{1}{4}}[/tex]
Finally, we are going to use the rule for fractional exponents: [tex]a^{\frac{1}{n} }=\sqrt[n]{a}[/tex]
Just like before, we can infer that [tex]a=2[/tex] and [tex]n=4[/tex], so let's use our rule:
[tex]2^{\frac{1}{4}}=\sqrt[4]{2}[/tex]
Or in words: the fourth root of 2.
