SAT Math Question of the Day

Math Category: Roots Level 2:

Q: Which of the following are true, if a > 1 and b > 1?

  1. √ a  * √ b  > √ (a * b)
  2. √ a  + √ b  > √ (a + b)
  3. √ a  + √ 2b  < √ (a + 2 b)

  1. I only
  2. II only
  3. III only
  4. II and III
  5. I, II, and III

UNANSWERED.

The correct answer is B.

A:

The answer is B.

√ a  * √ b  = √ (a * b) , so statement I is always false.

Now let's look at the second statement
√ a  + √ b  > √ (a+b) .
Roots are always positive numbers (the SAT does not deal with imaginary numbers, such as 2i), so if you square both sides, then the same > inequality still holds.
(√ a  + √ b )2  > (√ a + b )2
(√ a )2 + 2 (√ a )(√ b ) + (√ b )2  > a + b
a + 2(√ a )(√ b )  + b > a + b
2(√ a )(√ b )  > 0
Since the root of a number is always positive, the above statement is always true. So, we just proved that II is always true.

Now, lets look at III. We'll do the same thing we did for statement II.
√ a  + √ 2b  < √ (a + 2 b)
First, square both sides.
(√ a ) 2 + 2 (√ a )(√ 2b )  + (√ 2b ) 2 < a + 2b
a + 2(√ a ) (√ 2b )  + 2b < a + 2b
2(√ a ) (√ 2b )  < 0
Since a > 1 and b > 1, this means that √ a and √ 2b are > 1. 2(√ a ) (√ 2b )  > 1, so statement III is false.