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Square Roots: Calculating a square root of a number means the reverse of exponentiation. In effect, we are asking the following question: what number do I have to square to arrive at the result of whatever the number it is we have to find the square root of. To find a square root of 25, we would ask: What number do I have to square to get 25? And the answer is 5 To find a square root of 36, we would ask: What number do I have to square to get 36? And the answer is 6. As long as we are asked to find square roots of numbers that are themselves the results of exponentiation of integers, the answers are easy. What if we look for a square root of 30? We know that 52 = 25 and the next integer, 6 squared is 36. We therefore know that our answer is going to between 5 and 6. The only thing we can do at this point is to estimate and verify our guess by multiplication. Let us say that we will estimate our answer to be 5.4.
Therefore we see that 5.4 is close but a little low. If we try 5.5:
And we can see that 5.5 is a little too high. However with our guess of 5.4, we were 0.84 too low (30 - 29.16 = 0.84) and with our guess of 5.5, we were 0.25 too high, therefore we can see that the hundredth value of our answer will have to be higher than .05 - let's try .07:
We are on the right track and we could go even further in terms of looking for the next value - the thousandth, but the above approximation will be close enough. We can also be asked to find a square root of an expression. (Most often, this expression will contain an unknown, but we will not explore that area here. You may wish to visit the advanced math section of our website to find out all about unknowns and equations.) You may be asked to find the square root of 12 + 6. Obviously, since the expression does not include an unknown, we can calculate the value of 12 + 6 and then find its square root:
The square root of 18 will be more than 4, since we know than 42 is 16 and less than 5, since we know that 52 is 25 and we know it will be much closer to 4 than to 5, since the difference between 18 and 16 is only 2 and the difference between 25 and 18 is 7. By going through our trial and error technique we have shown earlier, we will arrive at 4.243 after rounding. Examples:
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