Question

Which expression is equivalent?

Answer 1

Third choice from the top is the one you want

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^{(1)/(4) }}{2^{(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^{(3)/(12) }}{2^{(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!!!