Addition and subtraction of roots can lead to common mistakes. Just like exponential expressions, you can’t just add the bases.

### Roots and Negative Numbers

Not all roots are possible. For example, √-4 is impossible since no number raised to an even power is negative. But odd roots of negative numbers are possible.

\sqrt[\displaystyle{3}]{-125} = -5 since (-5)^{3} = -125

\sqrt[\displaystyle{5}]{-32} = -2 since (-2)^{5} = -32

When taking an *odd root*, there is only one answer.

When taking an *even root*, there will be two answers.

For example, the square root of 49 is both 7 and -7. Both 7 and -7 squared are 49, so both are square roots of 49.

__But__ the GRE will only use the positive root for even roots. For √49, the only answer the GRE will use is 7.

Be careful when dealing with equations. Remember that an even root is undefined for negative numbers. A root must be positive, so

√(*x*^{2}) = |*x*| (NOT √(*x*^{2}) = *x*)

##### Example

If *x*^{2} = 4, what is *x*?

### Estimating Roots

Not all roots yield an integer. For example, √2 and √48 do not have integer values.

Using a calculator, you get √2 ≈ 1.414213562 … (The symbol ≈ means “approximately equal to”.)

It is usually enough to just simplify expressions by factoring rather than multiplying to get a value.

162 = √(2 × 81) = 9√2

If the answer choices are numbers, you can calculate to get an estimated value. How exact the answer choices are will tell you how accurate an estimate you need to make.

One way to estimate a value for a square root is to find the closest square root.

√48 will be slightly less than 7 since √49 = 7

To get a more exact value, use decimal approximations of common roots.

First, simplify by factoring. √48 = √(16 × 3) = 4√3

Then for √3, use an approximation. 4√3 ≈ 4(1.7) = 6.8

Knowing the approximate decimal value for these three roots will help when making estimates.

## Common Values

These roots commonly appear on the GRE, so you should be familiar with them. Since roots and exponents are connected, knowing the powers also tells you the value of the root. (The list of common values of powers is in the previous section, Exponents.)

### Advanced topics

*These topics are also in Chapter 6 Algebra*

How rough can estimates be? Can we use √3 = 1.7 or should we take √3 = 1.73? Can we estimate √60 as √64?

GRE questions are usually designed such that if you simplify an expression first and then make the closest well-known approximation possible, your answer should be ok. Avoid multiplication of estimated values, which amplifies the approximation error. It’s safer to add/subtract estimates.

But there are some tips to help you consider what approximation should be used:

#### 1. You can analyze the approximation error.

The approximation error when we have a number *x* rounded to some decimal is half of that decimal unit. If we round to the tenths digit the approximation error is ±0.05. What does this mean?

Let’s say some decimal is rounded to 1.4. It means that the real value lies somewhere in between 1.35 and 1.45. E.g. the numbers 1.4451, 1.36, 1.401, etc. are all rounded to the same tenth digit: 1.4.

So if we round to an integer some number *x* which has already been rounded to 1.4, we can be sure the original number will be rounded to 1. Because any number between 1.35 and 1.45 will be rounded to 1.

But what if we round 20*x* and use an approximation of *x* ≈ 1.4? The approximation error for 20*x* is 20 times greater than for *x*. We know that *x* lies somewhere in between 1.35 and 1.45 (1.35 ≤ x < 1.45), therefore 20*x* lies in between 27 and 29 (27 ≤ 20x < 29). So we don’t know if it rounds to 27, 28 or 29.

So if a question asks “20√2 is closest to” and the answer choices are 28, 31, 35, 36, 37, then 28 is the correct answer. But if the question asks “20√2 is closest to” and the answer choices are 28, 29, 30, 31, 32, then we must use a closer approximation for √2. Its approximation to the thousandths digit is easy to remember: √2 ≈ 1.414. But in this case, an approximation to 1.41 is enough.

The approximation error in this case is 0.005, so 1.405 ≤√2 < 1.415.

28.1 ≤ 20√2 < 28.3.

The correct answer is choice 28.

##### Radicals in Denominators

Terms in simplest form do not have a radical in the denominator.

##### Example

Simplify:

(a) \dfrac{4}{\sqrt{3}}

(b) \sqrt{\dfrac{5}{2}}

(c) \dfrac{6}{3\, – \sqrt{5}}

##### Radicals of Variables

When we deal with equations that contain variables under radical signs with an even degree (usually square radicals), we must keep in mind that such radicals can be calculated only for non-negative values. We also must keep in mind that such radicals yield a non-negative value.

There are two ways to deal with this issue:

- Plug the solution values into the original equation (or at least into the radicals in the original equation).
- Define the range of possible values for the variable right from the beginning.