If k^2 = m^2, which of the following must be true?

|k| = |m| suffice the condition of k^2 = m^2.

so ans: E

k^2=m^2
Taking the non negative square root of both sides of the equation ,
|k| = |m|
Answer E

With Even powers of variables the signs can't be predicted about them

Hence, We can't say anything about the sign of k and m being positive or negative
therefore, Option A, B, C and D are ruled out as all these options are hinting towards specific and known sign of k and/or m

For any sign of k and m, their absolute values must be same because k^2 = m^2, therefore, |k| = |m|

Answer: Option E

Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.


If k^2 = m^2, which of the following must be true?

(A) k = m
(B) k = −m
(C) k = |m|
(D) k = −|m|
(E) |k| = |m|



Since k^2=m^2 we have 0=k^2 – m^2 =(k-m)*(k+m). So k=m or k=-m.
So only (A) and only (B) cannot be an answer.
The choice (C) tells us that k should be greater than or equal to 0.
Similarly the choice (D) tells us that k should be less than or equal to 0.
So neither (C) nor (D) cannot be the answer.

The answer is, therefore, (E).

As squaring hides the sign of a number, k^2 will equal m^2 for every possible positive/negative combination of the two values.

The only statement we can make for sure is therefore (E) |k| = |m|.

Attached is a visual that should help.

Hello Bunuel

I inferred that square or even power of a number is always positive. Therefore k^2 and m^2 are positive. So |k|=|m|, no matter what the signs are.
I don't see why in the official guide its mentioned that we need to take root of k^2 and m^2 when we know that ^2 always gives positive number.

Do not follow what you mean bu the point is that \(\sqrt{x^2}=|x|\), so if we take the square root from k^2 = m^2, we'll get |k| = |m|.

Hi. Sorry for being not so clear

I meant to say that even powers ^2,^4, etc. always gives positive outcome no matter what the sign of the base is. For example:
(-2)^2= 4
and
(2)^2= 4

So what I inferred is that k^2 and m^2 are positive (no matter what the sign of k and m are). Therefore we can say that |k|=|m|.

Is this good?

Apart from knowing that the even roots and absolute values give non-negative result, we should also deduce that from k^2 = m^2 we can get |k| = |m|. Else, what would you say if one of the options were k^4 = |m|? Here both sides are also non-negative, but can we say from k^2 = m^2 that k^4 = |m| is true? No.

What does \(k^2 = m^2\) imply?

Only that |k| = |m|

(A) k = m
Not necessary e.g. k = 5, m = -5

(B) k = −m
Not necessary e.g. k = 5, m = 5

(C) k = |m|
Not necessary e.g. k = -5, m = 5

(D) k = −|m|
Not necessary e.g. k = 5, m = 5

(E) |k| = |m|
Always necessary. Note that absolute values of the two most be equal for the square to be equal. We cannot say anything about their signs though.

Answer (E)

Using the rule that \(\sqrt{x^2}=|x|\), the equation can be changed to:

\(\sqrt{k^2} = \sqrt{m^2} \rightarrow |k| = |m|\)

Answer E

The question asks us what MUST be true. So, if we can find a case where a statement is not true, we can eliminate that answer choice.

So, for example, one solution to the equation (k² = m²) is k = 1 and m = 1
Now let's check the answer choices.
A. k = m. Test: 1 = 1. Works. Keep A.
B. k = -m. Test: 1 = -1. DOESN'T WORK. ELIMINATE B.
C. k = |m|. Test: 1 = |1|. Works. Keep C.
D. k = - |m|. Test: 1 = -|1|. DOESN'T WORK. ELIMINATE D.
E. |k| = |m|. Test: |1| = |1|. Works. Keep E.

Okay, so the correct answer is A, C or E

Let's try another case. Another solution to the equation (k² = m²) is k = -1 and m = 1
Now let's check the remaining answer choices.
A. k = m. Test: -1 = 1. DOESN'T WORK. ELIMINATE A.
C. k = |m|. Test: -1 = |1|. DOESN'T WORK. ELIMINATE C.
E. |k| = |m|. Test: |-1| = |1|. Works. Keep E.

By the process of elimination, the correct answer is E

Cheers,
Brent

If K squared equals an M squared, that means that a K is equal to plus or minus M.

That basically solves the question, because K is not always M;

it's not always minus M; it's not equal to the absolute value of M because K might

be a negative, so negative cannot be something that's positive.

D says that K, which can either be positive or negative, is always equal to

minus of a positive number; well that's not always correct,

and E is the correct one because it says that plus a equals a plus.

Given: k^2 = m^2

Asked: Which of the following must be true?

If k^2 = m^2
Taking root on both sides
|k| = |m|

IMO E

As \(k^2 = m^2\) which means that one positive equals the other positive i.e.
+ = +

So we have two possibilities to show that '+ = +'
1. Using even powers : \((+/-)^2 = (+/-)^2 \)
2. Using Mods : |k| = |m| (since \(\sqrt{k^2}\) = |k|

Among the options only E satisfies. Other options involve one or the other variable which can take either positive or negative values. Here we need only positive terms.

Answer E.

Answer: Option E

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