Question

PLEASE HELP 50 POINTS

Answer 1

Step-by-step explanation:

The numerator of the exponent becomes an exponent. The denominator of the exponent becomes the index of the root.

$2^{\frac{1}{2}} = \sqrt{2}$

$2^{\frac{2}{3}} = \sqrt[3]{2^2}$

$3^{\frac{3}{2}} = \sqrt{3^3}$

$3^{\frac{1}{3}} = \sqrt[3]{3}$

Answer 2

Answer:

$2^{\frac{1}{2}}=\sqrt{2}$

$2^{\frac{2}{3}}=\sqrt[3]{2^{2}}$

$3^{\frac{3}{2}}=\sqrt{3^{3}}$

$3^{\frac{1}{3}}=\sqrt[3]{3}$

Step-by-step explanation:

First, we need to make some definitions about radical forms.

The expression $a^{\frac{b}{c}}$ is equivalent to :

$a^{\frac{b}{c}}=\sqrt[c]{a^{b}}$ (I)

Now let's work with the expressions given :

The first one is $2^{\frac{1}{2}}$

Using the expression (I) :

$2^{\frac{1}{2}}=\sqrt[2]{2^{1}}=\sqrt{2}$

The second one is $2^{\frac{2}{3}}$

Using (I) :

$2^{\frac{2}{3}}=\sqrt[3]{2^{2}}$

The third expression is $3^{\frac{3}{2}}$

Using the expression (I) :

$3^{\frac{3}{2}}=\sqrt[2]{3^{3}}=\sqrt{3^{3}}$

And the last expression is $3^{\frac{1}{3}}$

Using the expression (I) :

$3^{\frac{1}{3}}=\sqrt[3]{3^{1}}=\sqrt[3]{3}$