# Lesson: Introduction to Square Roots

## Comment on Introduction to Square Roots

Is x > y ?
(1) ✓x > y
(2) x² > y ### Tricky question! Just curious

Tricky question! Just curious, what's the source?

TARGET QUESTION: Is x > y?

STATEMENT 1: ✓x > y
There are several values x and y that satisfy statement 1. Here are two:
CASE A: x = 9 and y = 2 (notice that ✓9 = 3, and 3 is greater than 2. In this case, x > y
CASE B: x = 1/4 and y = 0.3 (notice that ✓(1/4) = 1/2, and 1/2 is greater than 0.3. In this case, x < y
Since we cannot answer the TARGET QUESTION with certainty, statement 2 is NOT sufficient.

STATEMENT 2: x² > y
There are several values x and y that satisfy statement 2. Here are two:
CASE A: x = 3 and y = 2 (notice that 3² = 9, and 9 is greater than 2. In this case, x > y
CASE B: x = 3 and y = 4 (notice that 3² = 9, and 9 is greater than 4. In this case, x < y
Since we cannot answer the TARGET QUESTION with certainty, statement 2 is NOT sufficient.

STATEMENTS 1 and 2 COMBINED
It's easy to show values of x and y that satisfy BOTH statements where x > y. For example, the following values of x and y satisfy BOTH statements:
x = 100 and y = 1
x = 16 and y = 1
x = 25 and y = -3
x = 1 and y = -30
etc
In all of the above cases, x > y
So, the correct answer MAY be C.

That said, the answer will E if there are values of x and y that satisfy BOTH statements AND where x < y

Is it possible to find such values for x and y?
No.

The tricky part is finding finding a situation where ✓x > y YET x < y

Important concept: in most cases, the square root of a number is less than the original number. For example:
✓9 < 9
✓4 < 4
✓36 < 36
The only time ✓x is GREATER THAN x is when 0 < x < 1. For example:
✓(1/4) > 1/4
✓(1/25) > 1/25
etc

So, if 0 < x < 1 then ✓x > x
The problem is that, if 0 < x < 1 then x² < x
So, any value on this range will satisfy statement 1, but will NOT satisfy statement 2.
As such, we CANNOT find values for x and y that satisfy BOTH statements such that x < y

So, it MUST be the case that, when both statements are combined, x > y

Thanks Brent :)

### Hi Brent,

Hi Brent,
https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-following-must-be-82080.html
Please let me know if it is already somewhere, if not, I would appreciate if you help and answer it, :
My answer is D. I and II only
There is too much debates there, it was confusing for me.
Thanks Here's my step-by-step solution: https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-followin...

Cheers,
Brent

### Thank you so much.

Thank you so much.

### Hi Brent, could you please

Hi Brent, could you please help me come up with a faster solution for the question below?

If x and y are positive, which of the following must be greater than 1x+y−−−−√1x+y?

I. x+y−−−−√2x+y2
II. x√+y√2x+y2
III. x√−y√x+yx−yx+y

A. I only
B. II only
C. III only
D. I and II only
E. None ### Hi Jalaj,

Hi Jalaj,

For example, in statement I (x+y−−−−√2x+y2), is the square root over BOTH 2x and y²?

Likewise, in statement II (x√+y√2x+y2), what is inside the first square root symbol? It looks empty.

If possible, it would be great if you could find the question online and send me the link.

By the way, what's the source of this question?

Cheers,
Brent

### Hi Brent, below is the source

Hi Brent, below is the source of the question. My apologies for the confusion.

https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-following-must-be-82080.html ### That helps a lot.

That helps a lot.
Here's my full solution: https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-followin...

Cheers,
Brent

### https://gmatclub.com/forum/is

https://gmatclub.com/forum/is-7x-1-2-an-integer-145680.html
sir can you please explain this? ### Hi Brent, do you agree with

Hi Brent, do you agree with the answer on GMAT club? How can the answer be statement 1 is sufficient?

Is 7x−−√7x an integer?
(1) x7−−√x7 is an integer
(2) 28x−−−√28x is an integer

(1) Atleast 2 values of x can satisfy this condition
x=7 and x=343.

How can the answer be statement 1 is sufficient?

Similary for statement 2 x=1/28 and x=7 satisfies the condition

so if we combine 2 statements we get x=7 since x=1/28 will not satisfy statement 1! ### I'm happy to help.

I'm happy to help.

ASIDE: In the future, please post a link to the question. I had a hard time finding the question (here it is https://gmatclub.com/forum/is-7x-1-2-an-integer-145680.html)

Here's my full solution: https://gmatclub.com/forum/is-7x-1-2-an-integer-145680.html#p2152171

Cheers,
Brent

Its a different approach. Can you please tell me whats wrong with my approach?

For statement 1. Atleast 2 values of x can satisfy this condition
x=7 and x=343...so how is this sufficient? TARGET QUESTION: Is √(7x) an integer?

STATEMENT 1: √(x/7) is an integer
You're right to say that x = 7 and x = 343 both satisfy statement 1.
HOWEVER, the target question doesn't ask us to determine the value of x; the target question asks us to determine whether √(7x) is an integer.

Let's test your two results (x = 7 and x = 343)
If x = 7, then √(7x) = √(49) = 7. So, the answer to the target question is "YES, √(7x) IS an integer".
If x = 343, then √(7x) = √(2401) = 49. So, the answer to the target question is "YES, √(7x) IS an integer".

There also are other x-values that satisfy statement 1.
For example, x = 28 also works. So does x = 63

Let's test these results as well
If x = 28, then √(7x) = √(196) = 14. So, the answer to the target question is "YES, √(7x) IS an integer".
If x = 63, then √(7x) = √(441) = 21. So, the answer to the target question is "YES, √(7x) IS an integer".

Notice that, even though there are infinitely many x-values that satisfy statement 1, when we test each of these possible x-values, the answer to the target question is ALWAYS the same: "YES, √(7x) IS an integer".
In other words, it MUST be the case that √(7x) IS an integer.

Since we can answer the target question with absolute certainty, statement 1 is sufficient.

KEY TAKEAWAY: Our goal is to determine whether each statement provides enough information to answer the target question with absolute certainty.

Does that help?

Cheers,
Brent

### Thanks Brent! So the target

Thanks Brent! So the target question should not satisfy different values. In this case its "is" or "is not" an integer. If the Q was find value of x so it was not sufficient.

I think with my approach would be time consuming since I have to keep searching values of x for which it will not satisfy. There is no limit to the search. Will stick to your approach. ### That's correct. IF the target

That's correct. IF the target question asked "What is the value of x?" then statement 1 would not be sufficient.

Here are two example to highlight the key concept:
--------------------
TARGET QUESTION: If x is an integer, is x ODD?
Statement 1: x is a prime number greater than 2.

Even though there are infinitely many prime numbers greater than 2 (e.g., 3, 5, 7, 11, 13, 17, 19, 23, etc), we know that ALL prime number greater than 2 are ODD.
So, the answer to the target question is "YES, x is most definitely odd"
Statement 1 is sufficient.
--------------------
TARGET QUESTION: If x is an integer, what is the value of x?
Statement 1: x is a prime number greater than 2.

In this case, we cannot answer the target question with certainty, since x can be 3 or 5 or 7 or 11 or 13 or . . .
Statement 1 is not sufficient.

Cheers,
Brent

### Thanks Brent

Thanks Brent
Sorry, one final time going back to the original question from GMAT club, this time for statement 2. the approach used is testing some values of x. But can we not use the same approach as we did for statement 1 which is solving the equation?
So it will go as:
let integer be p
so √(28x)=p
square both sides
28x=p²
7.4x=p²
7x=(p²)/4
take square root both sides
√(7x)=p/2=0.5p...not an integer

Not sufficient? Great idea!
I'd make just one small change to your approach:

At the end, once we know that √(7x) = p/2, we can't conclude that p/2 is NOT an integer.
For example, if p = 14, then p/2 IS an integer.
Conversely, if p = 3, then p/2 is NOT an integer.

Now that we have two contradictory answers to the target question, we can be certain that statement 2 is not sufficient.

Cheers,
Brent

### Oh yes, thanks for the heads

Oh yes, thanks for the heads up!
Not sufficient implies not getting a unique solution! we need atleast one "not" and atleast one "is" an integer Exactly!

### Hi Brent,

Hi Brent,

say we have
1/x= 1/(x+15)+ 1/(x-4)

The on solving this we get
x^2= -60
Does this mean there is no solution for x ### That's correct; the equation

That's correct; the equation x² = -60 has no real solutions.

Cheers,
Brent

### Hi Brent,

Hi Brent,

https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-following-must-be-82080.html

if I take x=9 and y=16 then statements 1 and 2 turn out to be greater. This is very confusing.

kindly help. thanks The keyword for this question is MUST be greater than...
You have demonstrated that the expressions in statements I and II COULD be greater than the original value.

Take a look at this solution: https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-followin...
It shows us that each statement need not be greater than the original value.

Cheers,
Brent

### Hey Brent,

Hey Brent,

at 0:35 you say that the square root of 49 can be 7 or -7. On the test day, however, it must be a positive number? I don´t know if I understood correctly. Because there are some DS questions where when the solution to an equation is the square root of a number, there are two possible solutions and thus, it is not sufficient?

Cheers,

Philipp ### The square root NOTATION (√)

The square root NOTATION (√) tells us to take the POSITIVE root of a value.
So, for example, √25 = 5 and √9 = 3

Conversely, if y² = 25, then y = 5 or y = -5

In other words, if y² = 25, then y = √25 or y = -√25

So, it all comes down to notation.

Does that help?

Cheers,
Brent

### I think I kind of understand.

I think I kind of understand. Now to take a specific example:

https://gmatclub.com/forum/is-x-an-integer-1-x-3-8-2-x-294299.html#p2265991

Here, if the question was not is x an integer, but is x a POSITIVE integer, would statement (2) be sufficient?

Cheers. If the target question were "Is x a POSITIVE integer?" then statement 2 would be sufficient.
We know this because √4 = 2 (and only 2)

Cheers,
Brent

### Hey Brent, another Q:

Hey Brent, another Q:

considering this:

https://gmatclub.com/forum/the-temperature-of-a-certain-cup-of-coffee-10-minutes-after-it-was-pou-98165.html

I get to 1=2*2^-10a

then I multiply by 2 to get 2^1=2^2*2^-10a

So i do: 1=-20a

Where did I go wrong?

Philipp This part of your solution is correct: 1 = 2*2^(-10a)
Multiplying both sides by 2 gives us: 2 = (2^2)[2^(-10a)]

Here comes the critical part: On the right side, we must use the Product Law and ADD the exponents.

When we do this, we get: 2 = 2^(-10a + 2)
In other words: 2^1 = 2^(-10a + 2)

We can now say that 1 = -10a + 2
.
.
.
a = 1/10 = 0.1

Cheers,
Brent

### Hi Brent,

Hi Brent,

For this question - https://gmatclub.com/forum/is-0-y-138931.html

For statement 2 if we take square root on both sides, we get +/- 1/2. However, as per your video, answer of square root of x should be |x|, which translates to only +1/2. Could you please clear when does the |x| rule apply?

Thanks! Be careful. That's not what I said.
The square root NOTATION tells us to take the POSITIVE root.
For example, √49 = 7 and √100 = 10

However, if we're told that x² = some positive number k, then EITHER x = √k OR x = -√k

### Oh, my bad. Now I get it.

Oh, my bad. Now I get it. Thanks!