Print out chapter

Definitions
Fractional Exponents
Roots and Negative Numbers
Estimating Roots
Simplifying Roots
Common Values 

Roots

For any positive integer n, the n^th root of a number x is r such that rⁿ = x.

If n is odd, then the n^th root of x is single-valued. E.g. the 3^rd root of -8 is -2, because (-2) × (-2) × (-2) = -8.

If n is even, then:

• The n^th root of x doesn’t have a value if x is negative, because even powers result in non-negative numbers. E.g. the 2^nd root of -4 doesn’t have a value, because there is no number r such that r² = -4.
• The n^th root of x has two values if x is positive, because any even power of the numbers r and -r is the same number.
  E.g. the 2nd root of 4 is 2 and -2, because 2² = 4 and (-2)² = 4.

A root is shown using a radical symbol √.  The number in front of the radical symbol is the degree of the root.  For square roots, the degree 2 is left off.

In order to avoid double-value of even roots, the radical sign with even degree is a non-negative value only.
For example, √4 = 2 (√4 ≠ -2), ⁴√81 = 3 (⁴√81 ≠ -3).

Note: On the GMAT, the radical symbol will be shown in two formats. The radical expression may not have a bar across the top. Expressions with no bars often use parentheses to group terms. For example:

Fractional Exponents
 
When a base is a non-negative number, fractional exponents are another way to indicate a root.  When the exponent is a fraction, the numerator says what power to raise the base to, and the denominator says what root to take.  You can raise the base to the power or take the root in either order.
 
Rules for simplifying roots are the same as for exponents – think of roots as exponential expressions with fractions as the exponents.  (Refer to the previous section for the rules of Exponents.)

Roots and Negative Numbers
 
Not all roots are possible.  For example, √-4 is impossible since no number to an even power is negative.  But odd roots of negative number are possible.
 
3√-125 = -5 since (-5)³ = -125.
 
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 GMAT will only use the positive root for even roots.  For √49, the only answer the GMAT 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 √(x²) = |x| (NOT √(x²) = x)

If x² = 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. 

         √2 ≈ 1.4          √3 ≈ 1.7          √5 ≈ 2.24

Round to the nearest integer:
 
(a) √3 + √2 
 
(b) √(25 – 10)
 
 

Simplifying Roots
 

Simplify: 

(a)  √468 

(b)  If a and b are positive, √(56a²b³)
 
 

Simplify, if x and y are positive:  (√xy )(√x³y )
 
 

Which expression is not equal to -3?

Common Values

These roots commonly appear on the GMAT, 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?

GMAT 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.

The value of √62 + 2√2 is closest to
 
A. 5
B. 11
C. 13
D. 16
E. 17 
 
 

The value of √60 × (√15 + 1) is closest to
 
 A. 38
 B. 40
 C. 44
 D. 46
 E. 48 
 
 

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 20x and use an approximation of x ≈ 1.4? The approximation error for 20x 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 20x 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.

2. You can analyze if your approximation increases or decreases the value.
 

2√60 is closest to which of the following?

A. 16
B. 17
C. 18
D. 19
E. 20
 
 

3. You can make two estimations: lower and upper.
 

Round 5/√17 to an integer.
 
 

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

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.
   

How many solutions are there for the equation √(x²– x – 6) = 1 – x  ?
 
 

Solve the equation:

(a) 3√x – 6 = 0

(b) 4√x – 5  + 6 = 18

(c) √9x – 20  = √4x + 5 
 
 

 

Contents of Number Properties Chapter: Table of Contents   Section 1: Number Rules   Section 2: Consecutive Numbers   Section 3: Divisibility   Section 4: Fractions   Section 5: Decimals   Section 6: Exponents   Section 7: Roots   Section 8: Sample Questions   Table of Contents

6. Exponents	8. Sample Questions