Even if you know the basic rules for questions involving powers and roots, it’s still common to feel some intimidation towards harder-looking GMAT questions.
The “” symbol is called the “radical” symbol. You may know the square root, but how comfortable are you with cube roots? For instance:
The “3” in the above is the “index” of the radical; to simplify radicals, use your knowledge of factors:
64 = 2 x 2 x 2 x 2 x 2 x 2
There are six 2’s in 64. Since the index is 3, we know we need three of the same number to pull it out from underneath the radical. Exponents and radicals appear most frequently on the GMAT in data sufficiency questions dealing with number properties:
If z < 0, then what is ?
If z is negative, let’s say -2 then –z = -(-2) = 2. Looking at the absolute value, the absolute value of z = -2 will be positive 2. Underneath the radical we would be left with 2 x 2 = 4. The square root of 4 is 2, which is the opposite of our original z = -2. No matter what negative value we plug in for z, the correct answer will always be -z.
When multiplying radicals, we can multiply the elements underneath the radicals:
?3 x ?2 = ?6
When dividing radicals, we can divide the elements underneath the radicals:
?6 / ?2 = ?3
When adding or subtracting similar radicals, treat the radicals like variables and combine like terms:
2?3 + 4?3 – ?2 + 7?2 = 6?3 + 6?2
On the GMAT, you will never be required to know a decimal equivalent of a radical. If you see a complicated-looking radical, try to “ballpark” it. For example, it is enough to recognize that ?110 is going to be slightly larger than 10, since ?100 = 10. Let’s look at a question involving exponents:
Is x2 > x ?
(1) x2 > 4
(2) x > -2
Because you know your exponent rules, you know that whether x2 > x is true is dependent on what kind of number x is.
From Statement (1), x > 2 or x < -2. For all numbers greater than 2 and less than -2, the answer to this yes/no question is “yes.” Sufficient
From Statement (2), let’s choose two values to try to get two different answers. If x = 0, then the answer to the questions would be “no,” however if x = 4 then the answer would be “yes.” A statement that allows us to answer a yes/no question with both a yes and a no cannot be sufficient.
The rules don’t change because powers and roots are combined with a secondary concept, such as inequalities here, so don’t let a combination of ideas throw you as you practice!
Vivian Kerr is a regular contributor to the Veritas Prep blog, providing tips and tricks to help students better prepare for the GMAT and the SAT.