If x<0 What is the value of [(x-3)^4]^1/4 + (-x|x|)^1/2 A. -3 B. x+3

Formatted the question.

Hi,

Let me understand that why [(-2-3)^4]^1/4 = 5. I think that [(-5)^4]^1/4 should be -5

There are 2 rules Here.

1. General Rule of Thumb - Every Number, Positive or Negative, raised to an Even Power will give a Positive Number.

Here is the Mathematics of same:

\((-10)^2 = (-10)*(-10) = (-1)*(-1)*(10)*(10) = (1) * (100) = 100\)

2. Rule followed in GMAT

If \(x^2 = 16\)
x = -4 or +4

But \(\sqrt{16} = +4\)

There are 2 rules Here.

1. General Rule of Thumb - Every Number, Positive or Negative, raised to an Even Power will give a Positive Number.

Here is the Mathematics of same:

\((-10)^2 = (-10)*(-10) = (-1)*(-1)*(10)*(10) = (1) * (100) = 100\)

2. Rule followed in GMAT

If \(x^2 = 16\)
x = -4 or +4

But \(\sqrt{16} = +4\)[/quote]
But still as it is -5^4*1/4
Simplifying it will give -5^1 = -5

I am so confused


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Hmm.. That is a valid point. I did not think of that. What I did was I solved the inside bracket first, applied the power 4 and took the 4th root. I don't know why your method is wrong though mine IS the right way to solve because otherwise we would have 2 different solutions of an operation.

I believe that one of the lesser known fallacies of Mathematics is at work here. For example, Here is One of those..

\(\sqrt{x} = \sqrt{(x)*(-1)*(-1)} = \sqrt{x}*\sqrt{(-1)}*\sqrt{(-1)} = - \sqrt{x}\)
but unless x=0, the above result cannot be true; however since we solved for a general term x, this must be universally true.

The fallacy lies in an operation that we did..

We cannot split \(\sqrt{(-1)*(-1)}\) into \(\sqrt{(-1)}*\sqrt{(-1)}\)

P.S. If you don't know, \((\sqrt{(-1)})^2 = (-1)\).

Two things to remember:

(1) \(\sqrt{x^2}=|x|\)

(2) When \(x \le 0\) then \(|x|=-x\), or more generally when \(\text{some expression} \le 0\) then \(|\text{some expression}| = -(\text{some expression})\). For example: \(|-5|=5=-(-5)\);

When \(x \ge 0\) then \(|x|=x\), or more generally when \(\text{some expression} \ge 0\) then \(|\text{some expression}| = \text{some expression}\). For example: \(|5|=5\).

Back to the question:

According to (1) \(\sqrt[4]{(x-3)^4}=|x-3|\). Now, since it's given that x is negative, then x - 3 will also be negative, thus according to (2) \(|x-3| = -(x-3)=3-x\).

Next, again since x is negative, then \(|x|=-x\). Thus, \(\sqrt{-x|x|}=\sqrt{(-x)(-x)}=\sqrt{x^2}=|x|=-x\).

So, finally we have \(\sqrt[4]{(x-3)^4} + \sqrt{-x|x|}=3-x-x=3-2x\).

Answer: C.

I am confused. How can you substitute x for a negative number, when x<0? wouldn't x have to be positive?

Hello Bunuel
When it's written |x|, don't we always take the positive value no matter what is inside. Why are we even considering the negative value of x when absolute value of a number is positive. Please explain.

Bunuel, abhimahna, BrushMyQuant

Absolute value rule is really creating confusion here. I read that absolute value of an integer/expression is always positive no matter what the sign is inside |x|. Then can't we just write here:

-(x-3)+ sq. root of (-x*x)
=-x+3+ sq. root of (-(x)^2)
= -x+3+(-(x)^2)^1/2
=-x+3+|-1|*x
= -x+3+x
=3

When I am removing | |, I am just considering positive value of whatever is inside | |.

Please help.

Hi Shiv,

You did everything right apart from opening the second part of the equation
sq. root of (-x*|x|)
Since x is negative so |x| = -x (since x is negative so -x will be positive)
sq. root of (-x*|x|) = sq. root of (-x*-x) = sq. root of (x^2)
Now whatever comes out of square root is always positive
So sq. root of (x^2) = |x|
Again, x is negative so |x| = -x (since x is negative so -x will be positive)
So final answer is
-x + 3 - x = 3-2x

Hope it helps!

Thank you so much for your reply.
May I ask a very basic question here?

We sat that absolute value of any number is positive. So shouldn't we write only positive value after removing the bars.
i.e. |x|= x

It's a very basic question but because of this only I am making mistakes in absolute value questions.

|x| = x if x is positive or zero
= -x if x is negative or zero

Example
|-3| = 3 = -(-3) (so |x| = -x if x is negative)

Sometimes, it becomes complex to solve questions when x is given as negative. So, to make things simple I suggest to take x = -k (where k is positive) and substitute x=-k in all the equations. Check my first reply in this thread to understand it better.

Hope it helps!

So what I understood is:

Absolute value of x is positive x when x is >=0.
Absolute value of x is negative x when x<0.

So we can't just say that absolute value of every number is positive no matter what is inside. The sign inside || will matter when || are removed.

|x| when x is positive= x
|x| when x is negative= -1*-x= +x

Your understanding is wrong. The post here: https://gmatclub.com/forum/if-x-0-what- ... l#p1847895 explains it.

Absolute value of a number CANNOT be negative! |x| is ALWAYS more than or equal to zero.

|some positive value| = that positive value

|some negative value| = -(that negative value), which still will be positive because -negative = positive.

|0| = 0.

Hope it's clear.

Can't we just cross off power 4 and 1/4?

x=-k

[(-k-3)^4]^1/4 + \sqrt{-(-k)*|-k|}
= -k-3+ k
= -3

I have tried this question so many times but still not really able to understand the minor things. I don't understand where I am going wrong: in cancelling the powers or opening the ||.

given x<0
so let x= -1
we get \(\sqrt[4]{(x-3)^4} + \sqrt{-x|x|}\)
this will be 5
substitute value of x =-1 in answer options
option C is correct

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