# When to Use Roots vs. When to Use Decimals on the GMAT

When to use roots and when to use decimals on GMAT questions can be a very tricky subject. Sometimes using roots will make the question work more smoothly, and other times using decimals will.

How can we decide? There’s not necessarily a good method to say “this is the right way,” because roots and decimals are basically the same thing.

## What about just turning them into actual numbers?

No way. Unless you have something that gives you an integer root, like $\sqrt{4}=2$ then your root will likely be an ugly, nasty number (“irrational” is the technical term) that doesn’t terminate or repeat.

While you will see lots and lots of roots/fractional exponents on the GMAT, you won’t be expected to translate them into “actual” irrational numbers.

## So what, really, is a root?

A square root of a particular number is another, often smaller, number that you’d have to square (multiply by itself) in order to get the original number back.

That is, the square root of 9 is 3 because if we square 3, we get 9.

$3^{2}=9$

Of course, that’s easy because we’re using integers. What happens if we pick something that’s not an integer?

Well, look at the square root of 2, for example.

$\sqrt{2}=1.4141$...

How can you calculate the square root of 2? Guess and check. Simple as that. These values have been guessed, checked, measured, and refined since Greek mathematicians were scribbling on parchment thousands of years ago.

They’re found listed as tables in books, on your computer, and in your calculator. In other words, roots are OBSERVED VALUES and not DERIVED VALUES.

You will never be expected to calculate observed values on the GMAT.

## Exceptions, Exceptions:

It’s worth noting that the square root of two is roughly 1.414.

$\sqrt{2}=1.414...$

It’s worth noting that the square root of three is roughly 1.728.

$\sqrt{3}=1.728...$

Moving along, let’s see how we can manipulate these numbers.

Take a case where we have a number to the sixth power and we take it again to the one-half power (in other words, we multiply it by itself six times, then take the square root of that).

$\left&space;(&space;x^{6}&space;\right&space;)^{1/2}=x^{6/2}=x^{3}$

Notice that this basically divides the powers.

## Pay attention to the relationship between roots and fractional powers.

While I’d suggest that the majority of the time on the GMAT it will be easier to think in terms of fractional powers, it is definitely sometimes easier to think about certain circumstances in terms of roots.

In this case, it would be easier to use fractional powers:

$\sqrt[3]{1.25*10^{-4}}$

The first thing I’m going to note here is the 1.25. If we were to express this as $1.25=125*10^{-2}$, then we’d be able to phrase the entire thing this way:

$\sqrt[3]{125*10^{-2}*10^{-4}}=\sqrt[3]{125*10^{-6}}$

Notice that, at this point, it’s probably easier to phrase it in the ⅓ power. That is:

$\left&space;(125*10^{-6}&space;\right&space;)^{1/3}=5*10^{-2}=0.05$

Here’s a case where it will be easier to use roots:

$\sqrt{\sqrt[3]{64}}$

Notice that we can look at the $\sqrt[3]{64}$ simply as 4 (because 4 is the cube root of 64). Then we see that it’s as simple as this:

$\sqrt{\sqrt[3]{64}}=\sqrt{4}=2$

It’s really easy to play around with this. Just take real, simple numbers such as 2 and 3 and spend some time rearranging their bases and powers to get a bit of practice.

$2^{3/2}=\left&space;(&space;2^{3}&space;\right&space;)^{1/2}=8^{1/2}=\sqrt{8}=2\sqrt{2}$

Then check that the numbers actually correspond to the correct numbers.

HINT: it’ll be simpler to do this if you avoid doing silly things like taking 2 to the ⅓ power--that will (obviously or not) give you some sort of ugly irrational number.

In short, learn how to switch back and forth between roots and fractional powers. It will make your GMAT experience a lot smoother.

Pro tip: look at the answer choices to determine how to deal with the root. You will ALMOST NEVER have to convert roots into decimals, especially when the answer choices are all given in the form of roots.

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