World Problem

[phoenix9801]

1st Question:

Can anyone tell me what is the difference when you have (x^2) vs (x)^2 ??????




2nd Question:

Also If plug in -4 or 4 into this equation it will give me the same result x^-2 < x^2 correct ????

[aneesh.kg]

1st Question:

Can anyone tell me what is the difference when you have (x^2) vs (x)^2 ??????

2nd Question:

Also If plug in -4 or 4 into this equation it will give me the same result x^-2 < x^2 correct ????

Answer 1: They just look different. Mathematically, (x^2) and (x)^2 are the same.
Answer 2: Whether you plug -4 or 4 in (x)^-2 < (x)^2, you will get the same result. The inequality holds true for both those values.

[phoenix9801]

Thank You, But I have one more question in mind, So how can I solve this problem using -4 and 4 ?????????


Which of the following statements must be true?

Indicate all such statements.

a) (square root(x))^3 = x^-1
b) (x^2) = (x)^2
c) x^-2 < x^2

1st Question:

Can anyone tell me what is the difference when you have (x^2) vs (x)^2 ??????

2nd Question:

Also If plug in -4 or 4 into this equation it will give me the same result x^-2 < x^2 correct ????

Answer 1: They just look different. Mathematically, (x^2) and (x)^2 are the same.
Answer 2: Whether you plug -4 or 4 in (x)^-2 < (x)^2, you will get the same result. The inequality holds true for both those values.

[vickythakre]

1st Question:

Can anyone tell me what is the difference when you have (x^2) vs (x)^2 ??????

2nd Question:

Also If plug in -4 or 4 into this equation it will give me the same result x^-2 < x^2 correct ????

Answer 1: They just look different. Mathematically, (x^2) and (x)^2 are the same.
Answer 2: Whether you plug -4 or 4 in (x)^-2 < (x)^2, you will get the same result. The inequality holds true for both those values.

Aneesh,

for second question.
x^-2 < x2

LHS : 1/x^2
Which will become 1/ (-4) ^ 2
i.e. 1 / -1 * (4^2)
i.e. 1/-16
RHS : (-4)^2
i.e. -1* (4)^2
i.e. -16

Now LHS = -0.0625
& RHS = -16
If i am not wrong then LHS > RHS.

Please correct me if i am wrong.

[aneesh.kg]

Aneesh,

for second question.
x^-2 < x2

LHS : 1/x^2
Which will become 1/ (-4) ^ 2
i.e. 1 / -1 * (4^2)
i.e. 1/-16
RHS : (-4)^2
i.e. -1* (4)^2
i.e. -16

Now LHS = -0.0625
& RHS = -16
If i am not wrong then LHS > RHS.

Please correct me if i am wrong.

Hi,

You've made a mistake there.

1/(-4)^2 = 1/(-4)*(-4) = 1/16
and
(-4)^2 = (-4)*(-4) = 16
are correct.

[aneesh.kg]

Which of the following statements must be true?

Indicate all such statements.

a) (square root(x))^3 = x^-1
b) (x^2) = (x)^2
c) x^-2 < x^2

a)
(x)^3/2 = 1/x
(x)^5/2 = 1
This works out only for x = 1 and not for any other value of x.

b)
LHS and RHS are the same thing.

c)
(1/x^2) < x^2
If you plug-in values, you will realise that the inequality does not hold true for values of x between -1 and 1 such as x = 1/2 and x = -1/2.

May I please know the source of these problems? They don't look like well-made GMAT problems, especially the second statement.

[phoenix9801]

Hi, Sure It a book for Standardized Test like Gmat and etc.. It also teach how Gmat can play with inequalities, word, problems, etc.... Main idea is to have a strong foundation. Thats is all. Let me know if you have any other questions.

Here is the actual Problem:

Which of the following statements must be true?

Indicate all such statements.

a) (square root(x))^3 = x^-1
b) (x^2) = (x)^2
c) x^-2 = x^2

I know for the Choice A is not correct because you can't have a negative inside a square root and the result will never be the same.

I know for The Choice B will always be True because of the cube power. Hence, It should be the only answer.


But When it come to choice C and plug in -4 and try with positive 4. it always comes true. Which become confusing. Maybe I am incorrect. If so can you show me how is it different when plugging in -4 and 4 step by step instructions please. If the statement is incorrect or correct. The way you explained it still confusing to me.

I really appreciate your help and time. If not too much to ask please explain it clearly to me. Thanks

Which of the following statements must be true?

Indicate all such statements.

a) (square root(x))^3 = x^-1
b) (x^2) = (x)^2
c) x^-2 < x^2

a)
(x)^3/2 = 1/x
(x)^5/2 = 1
This works out only for x = 1 and not for any other value of x.

b)
LHS and RHS are the same thing.

c)
(1/x^2) < x^2
If you plug-in values, you will realise that the inequality does not hold true for values of x between -1 and 1 such as x = 1/2 and x = -1/2.

May I please know the source of these problems? They don't look like well-made GMAT problems, especially the second statement.

[neelgandham]

a) (square root(x))^3 = x^-1
b) (x^2) = (x)^2
c) x^-2 = x^2

I will try to help you with this!

a) (square root(x))^3 = x^-1

(√x)^3 = x^-1
We know that √x = x^(1/2)
So, (x^(1/2))^3 = x^-1.
So, x^(3/2) = x^-1 [Because(x^a)^b) = x^(ab)]
If x = 0 then 0^(3/2) = 0^-1 => 0 = 0. So, (√x)^3 = x^-1.
If x = 4 then 4^(3/2) = 4^-1 => 8 = 1/4. No! So, (√x)^3 is not always equal to x^-1.
Statement A doesn't satisfy the 'MUST BE TRUE' condition.

b) (x^2) = (x)^2

(x^2) = (x)^2
(x)^2 = (x)^2. In simpler terms, both the terms mean the same.
If x = 0 then (0^2) = (0)^2 => 0 = 0. So, (x^2) = (x)^2.
If x = 4 then (4^2) = (4)^2 => 16 = 16.(x^2) = (x)^2.
Statement B satisfies the 'MUST BE TRUE' condition.

c) x^-2 = x^2

x^-2 = x^2
1/(x^2) = x^2[Because, x^-a = (1/x)^a]
1 = (x^2) * (x^2)
1 = (x^4) [Because, (x^a) * (x^b) = x^(a+b)]
If x = 1 then 1 = (1^4) => 1 = 1. So, x^-2 = x^2.
If x = 4 then 1 = (4^4) => 1 = 256. So, x^-2 is not equal to x^2.
If x = -4 then 1 = (-4^4) => 1 = 256. So, x^-2 is not equal to x^2.
Statement C doesn't satisfy the 'MUST BE TRUE' condition.

I hope that answers your question.

x^-2 = x^2

If x = 4,
then x^-2 = 4^-2 = (1/4)^2 = 1/16 and
then x^2 = 4^2 = 16
So the value of x^-2 is definitely not equal to x^2.

If x = -4,
then x^-2 = (-4)^-2 = (1/-4)^2 = 1/16 and
then x^2 = (-4)^2 = 16
So the value of x^-2 is definitely not equal to x^2

[aneesh.kg]

You're right. As I discussed in the posts above, the (c) statement works for both 4 and -4. However, the question is asking us if this statement 'must be true'. So it is not enough for the statement to work out only for a few values. Only if it works out for all the values of x can we say that it 'must be true'.

(x)^-2 < (x)^2 does not hold true for x = 1/2 and x = - 1/2:
(1/2)^-2 = 1/(1/2)^2 =(2)^2 = 4 > (1/2)^2 = 1/4
and
(-1/2)^-2 = 1/(-1/2)^2 = (-2)^2 = 4 > (1/2)^2 = 1/4
And this is enough for us to rule this statement out.

Does that clarify your doubt?