ETS Arithmatic problem(Roots Problem)

GRE Quant Problem Solving
1
  1. (-16)^1/4=?
  2. (-8)^1/4=?
    about this i read in ets math review that
    “For even order roots, there are exactly two roots for every positive number n and no roots for any negative number.”
    can you explain this by examples.
2

WolframAlpha is a great place to check these type of expressions

\sqrt[4]{(-1)(16)} = 2\sqrt[4]{-1}

\sqrt[4]{(-1)(8)} = \sqrt[4]{(-1)(2^3)} = \large2^{\frac{3}{4}}\sqrt[4]{-1}

Roots of a negative number are imaginary and are out of scope as per the GRE Math Review.

For example, 5 is a square root of 25 because 5^2 = 25.
Another square root of 25 is −5 because (−5)^2 is also equal to 25.

1 Like
3

For example, 8 has exactly one cube root, 8^1/3=2 but 8 has two fourth roots, 8^1/4 and
-8^1/4 whereas -8 has exactly one cube root, -8^1/3=-2 but -8 has no fourth root, since
it is negative.

4

ok i understand now
thanks for the help

5

Actually, -5 is not the square root of 25. Because sqrt(x^2) = modulus(x). The square root of a positive number is always positive and it can’t be negative. Correct me if there is anything incorrect.

6

no -5 is square root of 25

7

according to my understanding, when we calculate root of something its should satisfy the eqution.
so here -5*-5=25= 5 * 5
(-16)^1/4=this will have no roots because we can’t get -16 after multiplying 4 no. like 2222=-2-2*-2*-2=16
another ex. for negative no. whose order will be odd (-8)^1/3=-2=here -2*-2*-2=-8

8

“For even order roots, there are exactly two roots for every positive number n and no roots for any negative number.”
this is from ets .
and here it’s say that if n is positive and root is in even order then root = 2
like 16^1/4=-2,2 two roots
25^1/2=-5,5 2 roots
but no root for when n is negative and root is in even order
like (-16)^1/4=no roots
(-25)^1/2=no roots
next defination is for odd order roots
the odd order root will have one root it will not change on the n is nagtive or positive.
like 8^1/3=2 one root
-8^1/3=-2 one root

9

I checked the ETS guide, now I got it -
If someone asks what are the square roots of 25 then those are 5 and -5.

But if someone asks what is √ 25, then it is just 5. As it is mentioned in the ETS guide as well, "The expression consisting of square root symbol placed over a nonnegative number denotes the nonnegative square root. And when I check Wolfram Alpha Graph it shows the following too, plot Sqrt[x] - Wolfram|Alpha

10

yes right