By Mike MᶜGarry on October 24, 2018 , UPDATED ON November 15, 2018, in Powers and Roots, Video Lessons

You are watching: Small number in front of square root

As you know if you are going through this math video series in order — in the last video we looked at square roots. Now in this video we will look at other roots in math — such as cube roots, etc.

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Now when we get into that region between 0 and 1, then things get a little bit different. And you may remember, things were different here. First of all, if we take a root, the roots are bigger than b, assuming that b is this fraction between 0 and 1. And the higher the order of the root, the higher it gets.

So n is a higher order root, it’s higher. The nth root of b is higher than the mth root of b. So suppose we are taking different roots, say, of two-fifths. Well, two-fifths is less than the square root of two-fifths. This is gonna be less than the cube root which is less than the fourth root, and so forth.

We continue this pattern. It turns out that all these roots remain less than 1. So they get bigger and bigger and bigger, but they never get as big as 1. And this pattern continues with all higher orders of roots. So even if we had to compare two very high roots, we could say, for example, we know that the 50th root of two-fifths, that has to be greater than two-fifths but it has to be less than the 75th root of two-fifths.

And the 75th root of two-fifths still has to be less than 1. So we should be able to figure out where these four terms fall in an inequality even though we can figure up the exact decimal values of those roots. One way to summarize this we could say, the higher the order of the root, the closer the result is to 1. That is, with numbers bigger than 1, taking higher order roots make it smaller and smaller, move it closer to 1.

Taking roots of numbers between 0 and 1 makes it bigger and they move up closer and closer to 1. So this is the pattern, everything gets closer to 1. And one of the reasons for that, of course, is that we can take any root of 1 and it equals 1. These properties are rarely tested, and only on the hardest problems on the test.

So once again, this is not something you are gonna see every time you sit down for the test. These are very rare problems.


In summary, unlike with square roots, we can take cube roots of both positive and negatives, that’s a big idea. In fact, we can take any even root of positives only not negatives, but we can take any odd root of any number on the number line.

That’s also a really big idea. Any root of 1 equals 1, and any root of 0 equals 0. All roots preserve the order of inequalities assuming all the numbers are positive. And the higher the order of a root, the closer the result is to 1. So again, the numbers larger than 1 when we take roots, they get smaller and move closer to 1.

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When we take roots of numbers that are between 0 and 1, they get bigger and they move closer to 1.