Question

Which expression is equivalent?

Answer 1

Answer:

Third choice from the top is the one you want

Step-by-step explanation:

This whole concept relies on the fact that if the index of a radical exactly matches the power under the radical, both the radical and the power cancel each other out.  For example:

$\sqrt[6]{x^6} =x$ and another example:

$\sqrt[12]{2^{12}}=2$

Let's take this step by step.  First we will rewrite both the numerator and the denominator in rational exponential equivalencies:

$\frac{\sqrt[4]{6} }{\sqrt[3]{2} }=\frac{6^{\frac{1}{4} }}{2^{\frac{1}{3} }}$

In order to do anything with this, we need to make the index (ie. the denominators of each of those rational exponents) the same number.  The LCM of 3 and 4 is 12.  So we rewrite as

$\frac{6^{\frac{3}{12} }}{2^{\frac{4}{12} }}$

Now we will put it back into radical form so we can rationalize the denominator:

$\frac{\sqrt[12]{6^3} }{\sqrt[12]{2^4} }$

In order to rationalize the denominator, we need the power on the 2 to be a 12.  Right now it's a 4, so we are "missing" 8.  The rule for multiplying like bases is that you add the exponents.  Therefore,

$2^4*2^8=2^{12}$

We will rationalize by multiplying in a unit multiplier equal to 1 in the form of

$\frac{\sqrt[12]{2^8} }{\sqrt[12]{2^8} }$

That looks like this:

$\frac{\sqrt[12]{6^3} }{\sqrt[12]{2^4} }*\frac{\sqrt[12]{2^8} }{\sqrt[12]{2^8} }$

This simplifies down to

$\frac{\sqrt[12]{216*256} }{\sqrt[12]{2^{12}} }$

Since the index and the power on the 2 are both 12, they cancel each other out leaving us with just a 2!  Doing the multiplication of those 2 numbers in the numerator gives us, as a final answer:

$\frac{\sqrt[12]{55296} }{2}$

Phew!!!