- (-16)^1/4=?
- (-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.

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.

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.

ok i understand now

thanks for the help

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.

no -5 is square root of 25

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 2*2*2*2=-2*-2*-2*-2=16

another ex. for negative no. whose order will be odd (-8)^1/3=-2=here -2*-2*-2=-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

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

yes right