Thanks....we have not yet learnt how to cube root, and I find this rather interesting; why would we have a problem that involves this when we haven't learnt it yet. I think we have to learn it by ourselves...how do you do this? Thanks!
There are in general two ways to solve such problems, one is exact and one is approximate.
Let's take as an example the problem of finding the length of the side of a square with an area of 2 square meters.
The exact answer is that \(\displaystyle the\ length\ = \sqrt{2}\ meters.\)
But though it is exact, it is not a very practical answer because your meter stick does not show you the square root of two.
The approximate answer is that \(\displaystyle the\ length\ \approx 1.414 \ meters = 1,414\ millimeters.\)
You can get the approximation using a scientific calculator or through some method of successive approximation.
The simplest method of successive approximation is this:
Square root of 2 must be greater than 1 and less than 2 so let's try 1.5. But 1.5 * 1.5 = 2.25 so it must be smaller than 1.5.
Let's try 1.25. Well 1.25 * 1.25 = 1.5625. So the square root of 2 must be considerably bigger than 1.25 and somewhat smaller than 1.5. Let's try 1.4. Well 1.4 * 1.4 = 1.96. Hmm, it must be a bit larger than 1.4. Let's try 1.41. Squared that gives me 1.9881. And so on. (There actually is a method for finding approximations to square roots and cube roots without using repeated approximations, but I think that it has not been taught since calculators became affordable.)
Sometimes however there is a third way, by prime factoring.
\(\displaystyle \sqrt[3]{343} = \sqrt[3]{7 * 49} = \sqrt[3]{7* 7 * 7} = 7.\)
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