Radicals with roots less than one.....?

Question

Could anyone tell me a manual (no calculator) way of finding roots that are less than one? For example, the cube root of 8 = 2, but the .333333333 root of 8 = 512.

Also, the .714 root of 8 = 18.4 . If anyone can give me any advice, I'd greatly appreciate it. I thank you all very much for your time!

Very sincerely,

Motron

Answer 1

Remember, the nth root of x is simply x^(1/n). This relationship is used to define fractional roots -- simply invert and exponentiate. For instance, the 1/3 root of 8 = 8^(1/(1/3)) = 8^3 = 512. And .714 root of 8 is 8^(1/.714) ≈ 18.4, as you mention.

Answer 2

1. These are not really very good ways of looking at these kinds of "roots," because a certain semantic sleight of hand is being performed. 2. Also, once you have a very awkward "fractional root" involved, the search for a "manual solution" is a snare and a delusion --- theysimply don't exist:

1. The problem lies in the fact that the term "cube root" really means "raise to the power 1/3"; in other words, the implied " 3 " in the word "cube" appears in the denominator of the "power" to be applied to the number. Thus,

8^(1/3) = 2 BECAUSE 2^3 = 8. One shows this as follows :

Because 2^3 = 8, that means that [2^3]^(1/3) = 2 = 8^(1/3).

Similarly, assuming that by .333333333 you mean to indicate 0.333... (recurring), the analogy to "cube root"meaning a power of 1/3, the power implied now is 1 / [0.333... (recurring)] = 3. Therefore the "0.333... (recurring) root" is the number raised to the power 3, hence:

The "0.333... (recurring) root" of 8 = 8^3 = 8 x 64 = 512.

2. Using this equivalence of the "nth root" being the number raised to the power "1 / n", the " 0.714 root " means "raise to the power 1 / 0.714", i.e. "raise to the power 1.40..." [I suspect that you may have meant the "0.7142857...recurring" or "5/7th root" which means "raise to the power EXACTLY 1.4"] Assuming that, you have to recognize that in general there's no way of working it out manually with such a fractional decimal power. In this particular case it's:

8^(7/5) = 18.37917368... .

This has confirmed what you declared about your particular "fractional roots < 1."

However, I don't really recommend that you continue to think in terms of such " 1/nth roots." It's bound to create some confusion among lesser mortals when you discuss such things with others. The conventional thing is to talk in general of "raising [something] to [an explicit] power 1/n," rather than referrring to it as "finding the 1/nth root of [something]."

Live long and prosper.

Answer 3

The nth root of x is x^(1/n), so the 1/3 root of 8 is 8^(1/(1/3)) or 8^3, which is 512.

It is not necessarily easy to calculate this for an given value less than 1.

Answer 4

As you see in the answers taking a root is equivalent to raising to a power. If you wanted to do this manually (without a calculator) you could use logarithms.