Question

Simplify the expression 3^2/ 3^1/4 to demonstrate the quotient of powers property. Show any intermittent stepsthat demonstrate how you arrived at the simplified answer.

Answer 1

We are given a quotinet of two power expressions to be used to demonstrate the quotient property of powers:

$\frac{3^2}{3^{\frac{1}{4}}}=3^2\cdot3^{-\frac{1}{4}}=3^{(\frac{8}{4}-\frac{1}{4})}=3^{\frac{7}{4}}$

ANother way of doing it is to represent 3^2 as 3 to the power 8/4 so as to have the same radical expression.

Recall that fractional exponents are associated with radicals, and in this case the power "1/4" represents the fourth root of the base "3". That is:

$3^{\frac{1}{4}}=\sqrt[4]{3}$

So we also write 3^2 with fourth root when we express that power "2 = 8/4":

$3^2=3^{\frac{8}{4}}=\sqrt[4]{3^8}$

So now, putting that quotient together we have:

$\frac{\sqrt[4]{3^8}}{\sqrt[4]{3}}=\sqrt[4]{\frac{3^8}{3}}=\sqrt[4]{3^7}=3^{\frac{7}{4}}$

So we see that we arrived at the same expression "3 to the power 7/4"

in both cases. One was using the subtraction of the powers as the new power for the base 3, and the other one was using the radical form of fractional powers.