Tricky Exponents!

Algebra Level 2

True or False:

For all real numbers $y$ ,

$\Large y = \sqrt{y^2}$

False Impossible to answer True

10 solutions

Akhil Bansal
Sep 25, 2015

$(y^2)^{\frac{1}{2}} = \sqrt{y^2} = |y|$ $|y| = \begin{cases} +y \quad , \quad y \geq 0 \\ -y \quad , \quad y < 0 \end{cases}$
Therefore given expressions are like terms for $y > 0$ but unlike when $y < 0$

Akhil Bansal You've missed the case when y=0 in the definition of |y|. So it would be more accurate to say that $|y| = \begin{cases} +y \quad , \quad y \geq 0 \\ -y \quad , \quad y < 0 \end{cases}$ Right?

Sandeep Bhardwaj - 5 years, 8 months ago

Thanks,edited :)

Akhil Bansal - 5 years, 8 months ago

yup that is what i thought

Aditya Mohan - 10 months, 1 week ago

0 is not positive therefore you cannot say that y >/=0 is positive y

Timothy Vu - 5 years, 8 months ago

I thought like terms meant the were had same degrees.

Vibhu Saksena - 5 years, 8 months ago

I thought like terms meant they had same degrees. I don't know why the edit icon is not working.

Vibhu Saksena - 5 years, 8 months ago

In calculus, square root of Y is assume to be positive and is not negative without any thing mention about sign. In that case, it's impossible to answer because sometimes true and sometimes false.

Xi Huang - 5 years, 8 months ago

The statement says it is true for all real numbers. Hence, because it isn't true for some real numbers, the answer is false.

Sahej Veer - 5 years, 8 months ago

I sort of remembering some one on Brilliant say that the square root of 9 is +3 or -3 and that the square root of 3 squared is 3. That makes square root of y squared "y", wouldn't it?

Oliver Daniel - 5 years, 8 months ago

Your first statement isn't true. Fractional exponents do not connote the same meaning as root symbols. By the other rules of exponents, you could multiply the exponents together to get 1, thus making the statement true. If the question was \sqrt(y^2) I would obviously agree with you. However, because you can multiply encapsulated exponents, the question is incorrect.

Nathaniel Hoffman - 5 years, 5 months ago

since y is written equal to square root you can not admit its negative value

Gaurav Agarwal - 3 years, 7 months ago

However the square root sign if used without +or- in front is to be assumed positive unless told otherwise. Ie question needs to be more pedantic as normal rules of maths is applied due to no specification.

Grant Carnell - 5 years, 8 months ago

Agreed. The standard rule I've seen over and over again is that the root symbol means only positive, real roots, while fractional exponents denote any roots. I think the answer to this particular question is therefore true, since you can multiply the exponents out to 1.

Nathaniel Hoffman - 5 years, 5 months ago

Wait a minute Take -2 as a real number for y. -2= sqr (-2)^2 =》 -2= sqr 4 =》 -2= + or -2

Vitor Vecchi - 5 years, 8 months ago

Dragos Săndulescu
Sep 30, 2015

Easy to understand. Whatever number with the exponent 2 is positive so if the number was negative initially, it won't be negative when using the exponent too. That's basically the other solution said in words.

Jeremy Karis
Sep 25, 2015

lets say y=-x

-x=-x

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

different right? so they are not like terms

Can you please explain that how did you conclude that (x^2)^1/2=x? @Jeremy Karis

Sandeep Bhardwaj - 5 years, 8 months ago

we have X^2^1/2 take only the exponents it would be 2 multiplied by 1/2, which equals to one so we have left x^1 which is X

Dragos Săndulescu - 5 years, 8 months ago

Arulx Z
Jul 19, 2016

$\sqrt{-1^2}=1\neq-1$

Vinayak Kaushal
Jun 29, 2019

if we put y greater than 0 then we will get the same answer But if we put y lesser than 0 then it will come in positive which is not true So the equation is false.

Raihan Akram
Jun 12, 2018

say y=-1 but, root of -1 square = 1 #Not Equal!

John Link
Jan 11, 2016

The question "Is the following true for all real numbers y?" requires an answer of "yes", "no", or "can't determine". The choices offered ("True", "False", and "Impossible to answer") are therefore inappropriate.

I agree as given the question "what is the square root of y squared?" I'd give the answer "+y or -y"

Shelley Birchenough - 4 years, 5 months ago

Mohammad Saleem
Oct 1, 2015

You could test out multiple cases: 0, 3, -1, etc. When you use -1, you find that the equation is not always true.

Kerry Lasiter Billingsley
Sep 30, 2015

Not negative numbers.

Sadasiva Panicker
Sep 30, 2015

If y less than 0 it won't work

Can someone explain this? As far as I understand, -2^2 =4 and the square roots of 4 are 2 and -2, so this should work for all real numbers. Positive and negative.

Rande Farrell - 5 years, 8 months ago

Square root of x always denotes the principal, or positive, square root. (Sorry, can't seem to make LaTeX work.) So while 4 can be said to have 2 square roots, the square root of 4 is (only) 2.

Pamela Barnes - 5 years, 8 months ago

on the left of the equation the y would be neg and on the right it would be positive (-2 does not = +2)

Joshua Coddington - 5 years, 8 months ago

Tricky Exponents!

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