Lesson: Introduction to Square Roots

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

Answer: C

Hi Aladdin,

Here's my step-by-step solution: https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-followin...

Cheers,
Brent

Hi Jalaj,

I'd love to help you, but some of your entries are confusing and/or ambiguous.

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.
Alternatively, please add some brackets and spaces to prevent ambiguity.

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

Cheers,
Brent

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

Cheers,
Brent

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)

The correct answer is A.

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

Cheers,
Brent

Question link: https://gmatclub.com/forum/is-7x-1-2-an-integer-145680.html

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

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

Question link: https://gmatclub.com/forum/is-7x-1-2-an-integer-145680.html

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

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

Cheers,
Brent

Question link: https://gmatclub.com/forum/if-x-and-y-are-positive-which-of-the-followin...

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

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

Question link: https://gmatclub.com/forum/the-temperature-of-a-certain-cup-of-coffee-10...

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

Question link: https://gmatclub.com/forum/is-0-y-138931.html

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

Question link: https://gmatclub.com/forum/if-r-0-345-s-0-345-2-and-t-root-0-345-which-o...

It may be useful to recognize that all values between 0 and 1 behave the same way when it comes to their squares and their square roots.
So 0.345 will behave the same as any other value between 0 and 1.
So, rather than test the decimal 0.345, we can test the decimal 0.25 (since it's also between 0 and 1)
You'll find that it's much easier to work with 0.25, since we can quickly find its square root and square.

Notice that 0.25 = 1/4
So, √(0.25) = √(1/4) = 1/2 = .5
Also (0.25)² = (1/4)² = 1/16 = 0.625

So, (0.25)² < 0.25 < √(0.25)

Does that help?