Quite confusing!

$\sqrt{x^2} = |x|$ , so $\frac{\sqrt{x^2}}{|x|} + 1 = 1 + 1 = 2$

The common misconception is that $\sqrt{x^2}$ can be a negative number, but the only time square root returns a negative is if you're using it to solve an equation.

For instance, we say $\sqrt{100} = 10$ , but if we had the equation $x^2 = 100$ , then $x = \pm 10$ .

This is the first question on brilliant where I don't agree with most of the solutions.

Disagree with this solution.

Answer is 2 or 0.

Mod(x) is always positive. Taking the positive root gives 2. Taking the negative root (regardless of what value x takes) the answer is 0.

The first term is just equal to 1, thus $1+1=2$

actually, the square root of x^2 can yield a negative value as well... so the answers are 2 and 0...

2 and 0 both are correct.

I think, |x| means just the magnitude of x (i.e if x is negative mod x will be positive) and thus the denominator will always be positive irrespective of numerator. Sqrt(x^2) = +x or -x => Sqrt(x^2)/|x| = +1 or -1, so this can have two solutions either 0 or 2.

|x|=x if x> or equal to 0 |x|= -x if x is less than 0 so if we take x as positive number then its root of its square will be positive as well as its mod will be positive and if we take x as negative number then root of its square is negative as well its mod negative so in both cases the answer will be 2

√(x^2) =x

so, x/x =1

and ,1+1 =2

the square root of x square is absolute x. absolute x over absolute x is 1 (both cut each other) 1 + 1 = 2

I don't understand

Two is not the only answer. Zero is also another possible answer. But when I entered 0 it didn't take it as correct answer!

doesnt actually work root of something can be either negative or positive you get either 2 or 0

sqrt(x^2)=|x| => |x|/|x|=1=> |x|/|x|+1=1+1=2

LOL WHY DO YOU GUYS DO HAVE SOLUTIONS FOR THIS? x is just any number. Plug in any number lol to x, and you will always get "2"

i think it should make it clear that x is positive, otherwise it could also be 0

The first term x/x or -x/-x which is just 1.

therefore:

1+1 = 2

as x is a non zero real no.the value of 1st term will be x/x=1 and thus 1+1=2

Root over of X^2 = |x|

by definition: $\sqrt{x^2} = |x|$ . Thus, the following question will be 1 + 1 = 2

Remember that $\sqrt{x^2} = |x|$ . $\begin{aligned} \frac{\sqrt{x^2}}{|x|} + 1 &= \frac{|x|}{|x|} + 1 \\ &= 1 + 1 \\ &= \boxed{2} \end{aligned}$

Therefore, the answer is 2 .

For any real number $x$ not equal to $0$ , there are two distinct complex numbers whose squares are equal to $x$ .

However, this does not mean that $\sqrt{x}$ has two values. By convention, the square root function is defined such that $\sqrt{x}$ is equal to the unique positive real number whose square is $x$ , whenever $x$ is a positive real number. Also, $\sqrt{0} = 0$ . The value of $\sqrt{x}$ is not defined when $x < 0$ (the domain of the function is the nonnegative real numbers).

In fact, this definition is precisely why the solution to $x^2 = y$ , say, is $x = \pm \sqrt{y}$ (when $y$ is a positive real number). Here, $\sqrt{y}$ is defined to be the unique positive root of $y$ . Thus, to list all solutions to the equation, we actually have two possibilities: $x = \sqrt{y}$ and $x = -\sqrt{y}$ . We write $x = \pm\sqrt{y}$ as shorthand for these two possibilities. This notation does not indicate that $\sqrt{y}$ has two values (the opposite, actually).

In this question, $x^2$ is guaranteed to be a positive real number, because $x$ is a non-zero real number. $\sqrt{x^2}$ is the unique positive number which squares to $x^2$ - in other words, $\sqrt{x^2}$ is equal to $|x|$ (uniquely).

So the expression should be simplified to

$\frac{\sqrt{x^2}}{|x|} + 1 = \frac{|x|}{|x|} + 1 = 1 + 1 = 2$

$2$ is the unique answer to this problem.

N.B. It is only true that $\sqrt{x^2} = x$ when $x\geq 0$ . In general, $\sqrt{x^2} = |x|$ for real $x$ , by definition. This definition is taken as a matter of convention in order to allow the definition of a square root function from $\mathbb{R}^+ \rightarrow \mathbb{R}$ at all (the value of a function must always be uniquely defined by its arguments).

X is not zero .then let x is 1 . Let I square + 1 = 1+1= 2. Therefore answer is 2.

What about imaginary number? Where i can be a negative number. So √i can be -1. Any advise? I was taught this way. Now I'm confused.

The square root of a number squared is just that number so √x^2 is just x. it must be positive because for example -2^2=4 then square that is just 2. In this equation x=1 so √1^1+1 is simply 1+1=2

but what if x = 0 ????/

x square root is the same thing as x. So if you take x as 10 so 10/10 gives you one so you will add another 1 to it and get the answer 10

If x=10 then 10/10+1 =1+1 =2 (the answer)

Agree, it has to be 0 or 2. Both are possible solutions.

I feel that this neglects the possible solution i + 1 and should say that x is any positive number

Since the square route of x^2 = |x|, then the square root of x^2/ |x| + 1 = 1 + 1=> 2

$\pm\sqrt{x}$ is the notation used when we want both the positive and negative root of a number.

$\sqrt{x}$ is the notation referring to the positive root.

ergo the answer is 2.

Why can't it be zero? if the number is negative it can be....

Case-I.

If x<0, then (x)^2= x^2.

So , √(x^2) = - x , as x< 0.

And | x| = - x ; as x<0.

Then 【 √(x^2) ÷ |x| 】+1 = (-x/-x) + 1 =1+1= 2

Case-II

If x>0 , then √[ x^2] = x, as x>0.

And |x| = x, as x>0.

Then 【 √(x^2) ÷ |x| 】+ 1 = (x/x) +1 = 1+1 = 2

Hence, in both the cases ; the answer is 2.

put the value x=1 and (1)/1+1=2 It is ao easy.....

Sqrt(x^2) = x. x/x = 1. 1 + 1 = 2 . It's as simple as that

I think the Only Answer is 2.....; even If x is -ve, x^2 is +ve. and then comes square root. so it is always 1+1 = 2

√x²=x .. x/x = 1 .. 1+1 = 2

Treat x as 1. The square root of 1 squared is 1. The absolute value of 1 is 1. 1/1 is 1. 1+1 = 2

Its positive when square root are given and so all the real numbers can set here and it will be positive

cancel square root and square it becomes (x/x) + 1 then ; let x = 1 ; and it becomes like this (1/1) + 1 = 1+1 = 2 that's it!

√x²÷x+1== x÷x + 1 ==→ 1+1==→2

(x^2)^1/2=+x or -x.

If it is +x then x>0, |x|=x.

If it is -x, then -x<0, |x|=-x.

Hence in both cases the answer is 2.

P.S.- we know that |x|= -x, for x<0 and +x for x>0.

I'll try to make this as simple as possible since some don't know how. Ok first the square root of x^2 is just x and anything divided by itself is 1, add 1 more and you get 2 very simple.

SInce it is given that X can not be zero I.e.x is a non zero.when X is cumng out of sqr root then negative value of X is rejected therefore we are now left with only positive value of X. \sqrt{X} divided by X gives us 1.so 1+1 = 2

suppose X=1

Even if x is negative, the answer would be 2.

If x is negative, the square would be positive. Then the square root would also be positive.

The square root of x^2 would be the equivalent of the absolute value of x, so dividing them would equal 1.

1+1=2.

My sir says that it's a misconception that √x^2 = +/- x. √x^2=+x and -√x^2 = -x.

I solved it like a normal fractional expression; set the denominator of +1 to 1, and solve from there. Keep in mind that the absolute value of x = the root of the square of x = x. you should eventually end up with something like 2x/x. Cross-out the x's and voila, you have 2 (sorry for the verbal description. I don't really know how to use LaTex)

I got the right answer of 2, but I don't understand the comma and no equals sign, what does that mean?

root x square /x+1 root and square will cancel and we will get x/x+1 , when we take LCM we get , x+x/x which gives 2x/x x and x gets cancelled and we get 2 so answer is 2

√x^2=x &lxl=(+x)or(-x) So-x/lxl+1=1+1=2 Or,x/lxl+1=(-1)+1=0 But 0is not in option so answer is 2

Say, a^2 = x Then: a=+Sqrt(x), - Sqrt(x)

Square Root sign, called radical, means principal/positive square root. E.g., 5 and -5 are two square roots of 25, but if we write 25 inside radical, it means 5 and not - 5. Use Plus Minus sign for all possible solutions.

Therefore, the answer is 2.

I just substituted 1 in for x because the square root of 1 is 1 and the absolute value of 1 is also 1 Then I simplified.

\sqrt{x^2}=\:x\:why\:\left|x\right|

The answer is 2, read the question, where X is a non zero real number :D

The solution can be either 0 or 2 as it depends on the value of x being non-zero. If x is negative then the solution is zero -1+1=0 and if it is positive then the solution will be that positive x and positive x will cancel each other out giving positive 1, hence 1+1=2

since sqrt(x) = mod(x)

Root Of x square is x , and absolute x is also x , therefore x/x is 1 ,and 1=1 is 2 . division of both number will always be 1 thas y the answer is 2

I'm really 16 and I guessed the answer 2

The crux of this problem is $\frac{\sqrt{x^2}}{ \left|x\right|}$

One piece at a time, simple first: $\left|x\right| = x$

Just to be clear: this is always positive because it is an absolute value. This is never -x.

Second piece: $\sqrt{x^2} = x$

This is also always positive because you are taking the sqrt of x^2 and x^2 is always positive. Since only positive x is used, then the only solution is 2 .

Note that if the equation used sqrt(x), then you would need to account for both the negative and positive solutions (i.e. -sqrt(x), sqrt(x)).

since x is a non real number i.e is x>0 hence answer is 2

Write a solution. "The square root of any number squared is that number itself. That number represented by X is then divided by absolute value of itself, which is1. THEREFORE 1+1=2.

Rather than complicating things, or even over-thinking the problem, I took to sheer basics. The stipulations for "x" were a non zero real number. Subsequently speaking the next real number following 0 is 1. By substituting 1 for "x," our finish product is in fact 2.

Proof: Numerator: The square root of 1 is 1. 1 squared is still 1. Denominator: The absolute value of any number is the positive number itself; |1| = 1. So we have 1/1. The result of dividing 1 by 1 is still 1. Finally we arrive to a simple equation, 1+1 and as we all know the result is 2.

Nice and simply, just substitute 1 for "x. "

(x^2)^1/2=mod x

akar x kuadrat=x x:x =1 1+1=2

square root of x squared is x. x divided by x is 1. 1 plus 1 is 2.

Karthik is correct !

The square root of x squared is just x, so you can reduce this equation to $\frac{x}{x} +1$ , which becomes 1 + 1, or 2.

Since absolute value is imposed for the denominator, the value of x, whether negative or positive, will end up positive. The numerator has been squared, so whether it's negative or positive, it will end up positive. The square root of a positive number has two solutions, a positive one and a negative one, so the final answer can either be 2 or zero. However, the equation does not require both roots (usually indicated by a +- in front of the equation, like in the quadratic formula), meaning we are to use the PRINCIPAL root, which is the positive number, so the answer is 2.

|x| has to be bigger than 0. anything sqrooted also has to be bigger than 0. therefore it its just x^2 which is sqrooted to x, then devide this by x, therefore 1 + 1 = 2

{(x^2)^0.5}/x + 1 = 2

Since Mod of X is always +ve quantity...Answer is 2

i'll write solution by ِِArabic: الجذر التربيعي لاكس أس 2 يساوي اكس اكس علي اكس يساوي واحد واحد+واحد يساوي 2

x/x+1=2x/x=2

Root of one number has always has two solutions i.e -ve and +ve. so even if x*x results in positive and after that root of that number results in two intezers i.e +ve and -ve and so it results in two answers one is 2 and the other is 0.

square and square root gets cancelled and denominator becomes "x" . therefore x/x=1 and 1 +1 = 2

sqrt of x^2 = x , so x/x +1 =2

x /x + 1 = 1 +1 = 2

Either you choose positive no. or negative no. , the first term in the expression becomes 1 , so answer is 1+1=2

Given ans. is 2. But considering +/- values of sq. root of x*x, the other ans. can be 0

1+1=2 The square root and squared functions basically cancel out, and since the denominator x is already positive, the absolute value doesn't matter

(sqrt(x))^2 = |x|. |x|/|x| = 1. 1+1 = 2

x^(1/2)=X x/x=1 1+1=2

Square the entire thing. √x2 becomes x2. |x| becomes x2. Squaring 1 yields 1. x2/x2 = 1 + 1 = 2 ans.

When the question says, "x is any real number," this usually implies that if you insert any number into the equation, then it will give you the answer. Just in case though, you should try inserting multiple different values for x. For example in this problem, if x equals 1 then the answer is 2, and if you say x equals 2 the answer is still 2.

IT MEANS (x/x)+1=1+1=2

i was first confused that if could be 0 too, but 2 is for sure. because, it's modulus of value.

The radical symbol by convention is set to equal the positive root, so the square root of x squared is always positive for any non-zero real number. Therefore absolute value of x and square root of x squared are always the same value and when divided equal one. Add one to get two, the final answer.

In mathematics |x| of real number of x is non-native value so √x^2 = x and |x| = x

so x/x+1 = 2

(X^1/2)(X^2) = x , X/x=1, 1+1=2

√X/│X│+1. Where x=1 →√1/│1│+1=2.

x^2 is always +ve . so square root of x^2 is +ve. Hence x/x+1=1+1=2. only ans is 2.

The correct answer is not just 2, it may be 2 or 0 as the sqrt of any number has two solutions +x and -x. the solution would be 2 only if we are dividing by x not |x|

if X is a non zero real number (X^2)^1/2=IxI, thus, the above equation will always be number 1+1=2

This has two solutions. 0 and 2. No argument.

always root of x square will be x. so the root and square get cancelled . Now the sum is x divided by x +1. so the x and x get cancelled and it becomes 1 and 1+1=2 so the answer is 2

(sqr root of x^{2}) =IxI
IxI/IxI=1
hence, the ans is 1+1=2

Because the "x" in the square root is being squared, it has to be positive. Same thing with the absolute value.

The square root and squared functions basically cancel out, and since the denominator x is already positive, the absolute value doesn't matter. These are just making everything more complicating. So now you have x/x, and any number divided by itself is one, so now you have 1+1, which is obviously 2.

I used the number one, because it is a non zero real number. Of course, it is a relatively easy number to use, that is why I chose it for explaining.

So, first I plugged in x as 1, and got the square root of one squared is 1, over the absolute value of 1+1.

That would equal 1/1+1, which is 1+1. 1+1=2, and two is a correct solution, as I have been told by the guide.

The square root notation (without sign) represents the POSITIVE square root. It follows that;

$\boxed{|x| = \sqrt{x^2}}$

which is sometimes used as a definition of absolute value.

Therefore;

$\frac{\sqrt{x^2}}{|x|}=\frac{\sqrt{x^2}}{\sqrt{x^2}}=\frac{|x|}{|x|}=1$

and the equation simply becomes;

$1+1=2.$

The answer should be 2 or 0 as square root of X will have both +X and -X as solution. Mod X will always be positive. So for +X in numerator answer is 2 and for -X its 0

Square always shows that a number is positive . x may be +ve or -ve For example:Let x = -5 or 5 The square of -5 and 5 both equals to 25 and square root of 25 = 5 and also |5| = |-5| = 5 => 5/5 +1 = 1+1 = 2 Answer

it can b either 2 or 0

It's Just simple , if u take x=2 than ^x wil be 4 and x = 2 , apply L.c.m = 2+2/2 Ans ! = 2

Another answer would be zero unless it's specified that we are taking the principal or positive square root. If we had (sqrt(x^2))^2 + 1 then our only answer would be two as it no longer matters if the square root is positive or negative in that case.

Because mod(X) is always a positive value there the first fraction is equal to 1 therefore the answer is 1+1=2

in this problem they ask non zero real numder so x=2.then root and power 2 chancl we get 2/2=1 now 1+1 =2 now ans is 2

Given: x is a non zero real number

Step 1: consider x as a least non zero real number. let say x= 1 Step 2: substitute x in the given equation. Step 3: solving it we get 1+1=2

Note: This result is true for all non zero real numbers.

the root of x squared is1 , the value of x is 1 so that 1+1=2

square root of every non zero number is always positive (Even the value of X is -ve). So its comes 2, every value of X

same as kartik solution . u should refer to graph plotiing of root x^2

As x is a real no. x1/2=|x|

And thus 1+1=2

When a variable such as x is squared, the value of x as a whole is squared, meaning x^2 will always be a positive number. The square root reduces result of x^2 to just x, while the bottom will always be positive x. Any value divided by itself will always be 1, so x/x = 1. And, as always, 1 + 1 = 2

square root and exponent can be cancelled and when you divide the number by itself, the answer is one, so then you'll end up with 1+1=2 :)

2 or 0.

If x>o, the ans is 2. If x<0, the ans is 0.

frnds mod of |x| or any number gives always +ve value and root of x^2 is x and it's also a +ve so on solving it gives answer 2..

Square and root which after a simple simplification gives us +ve "X" and modules of x is "X" . So its gives us 1 and 1+1=2

(x)^2=(-x)^2
But Root of (x)^2 = x
While Root of (-x)^2 = -x
Hence Here, Root of (x)^2 = x
|x| = x
Hence, 1+1=2

Square root of x^2=x then, |x|=+-x but x is a non zero real number. Because, |x|=x then, x/x=1 1+1=2.