Lesson: Fractional Exponents

Comment on Fractional Exponents

Please help me with this one.

Is 3^(a²/b) < 1 ?
(1) a < 0
(2) b < 0
gmat-admin's picture

TARGET QUESTION: Is 3^(a²/b) < 1 ?

This is a great candidate for REPHRASING the target question. Notice that, if order for 3^k < 1, it must be the case that k < 0.

So, in order for 3^(a²/b) < 1, it must be the case that a²/b < 0 (i.e,, a²/b must be negative). So, let's REPHRASE our target question as....

REPHRASED TARGET QUESTION: Is a²/b < 0?

STATEMENT 1: (1) a < 0
There are several values a and b that satisfy statement 1. Here are two:
CASE A: a = -1 and b = -1. In this case a²/b = (-1)²/(-1) = -1. Here, a²/b < 0
CASE B: a = -1 and b = 1. In this case a²/b = (-1)²/(1) = 1. Here, a²/b > 0
Since we cannot answer the REPHRASED TARGET QUESTION with certainty,
statement 1 is NOT sufficient.

STATEMENT 2: b < 0
There are several values a and b that satisfy statement 2. Here are two:
CASE A: a = 1 and b = -1. In this case a²/b = (1)²/(-1) = -1. Here, a²/b < 0
CASE B: a = 0 and b = -1. In this case a²/b = (0)²/(-1) = 0. Here, a²/b = 0 (in other words, a²/b is NOT less than 0)
Since we cannot answer the REPHRASED TARGET QUESTION with certainty,
statement 2 is NOT sufficient.

STATEMENTS 1 and 2 COMBINED
If a < 0, then a CANNOT equal 0, which means a² is POSITIVE
If b < 0, then b is NEGATIVE
So, a²/b = POSITIVE/NEGATIVE = NEGATIVE
In other words, a²/b < 0
Since we can answer the REPHRASED TARGET QUESTION with certainty, the combined statements are sufficient.
Answer: C

I memorized all the square roots as you discussed in a previous video. However, I am having a hard time trying to quickly calculate the cube root (or 4th etc.)

Is there an easy way to quickly know this? Or another list that should be memorized for the GMAT?
gmat-admin's picture

For cube roots, you must recall the powers with exponent 3 (2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, etc)
So, for example, ∛64 = 4

For fourth roots, you must recall the powers with exponent 4

Cheers,
Brent

Thanks for the comment! I am sure with more practice questions, it will start becoming quicker for me to recall.

Hi Brent! Glad to post again !

Regarding the following question

https://gmatclub.com/forum/if-m-4-1-2-4-1-3-4-1-4-then-the-value-of-m-is-93340.html

I understood the value of M like: 4^1/2 + 4^1/3 + 4^1/4. Looking for a common denominator I ended up with 4^13/12, which is a little bit more than 1 so, the value of M would be something more that 4. Hence, answer E. Is that approach correct?

warm regards,
Alejandro
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-m-4-1-2-4-1-3-4-1-4-then-the-value-of-m-is...

Even though you arrived at the correct answer, it was purely by coincidence.
The problem is that we can't add exponents the way you have. We can only add exponents when we're multiplying powers.
For example, (7^5)(7^10) = 7^15

When it comes to adding, however, there are no convenient simplifications.
For examples we can't say that 7^1 + 7^2 = 7^3

Cheers,
Brent

Thanks for your words

The beeth root... Love it!

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