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Powers and rootsIntroductionPowers are used when we want to multiply a number by itself repeatedly.PowersWhen we wish to multiply a number by itself we usepowers,orindicesas they are also called.For example, the quantity 7×7×7×7 is usually written as 74. The number 4 tells us the numberof sevens to be multiplied together. In this example, the power, or index, is 4. The number 7 iscalled thebase.Example62=6×6 = 36. We say that ‘6 squared is 36’, or ‘6 to the power 2 is 36’.25=2×2×2×2×2 = 32. We say that ‘2 to the power 5 is 32’.Your calculator will be pre-programmed to evaluate powers. Most calculators have a button markedxy, or alternatively ˆ . Ensure that you are usin gyour calculator correctly by verifyin gthat311= 177147.Square rootsWhen 5 is squared we obtain 25. That is 52= 25.The reverse of this process is calledfindingasquareroot. The square root of 25 is 5. This iswritten as2√25 = 5, or simply√25 = 5.Note also that when−5 is squared we again obtain 25, that is (−5)2= 25. This means that 25 hasanother square root,−5.In general, a square root of a number is a number which when squared gives the original number.There are always two square roots of any positive number, one positive and one negative. However,negative numbers do not possess any square roots.Most calculators have a square root button, probably marked√. Check that you can use yourcalculator correctly by verifyin gthat√79=8.8882, to four decimal places. Your calculator willonly give the positive square root but you should be aware that the second, negative square root is−8.8882.An important result is that the square root of a product of two numbers is equal to the product ofthe square roots of the two numbers. For example√16×25 =√16×√25=4×5=20More generally,√ab=√a×√bbusinesswww.mathcentre.ac.ukcmathcentreMay29,2003
However your attention is drawn to a common error. It is not true that√a+b=√a+√b.Substitute some simple values for yourself to see that this cannot be right.Exercises1. Without usin ga calculator write down the value of√9×36.2. Find the square of the following: a)√2,b)√12.3. Show that the square of 5√2is50.Answers1. 18, (and also−18). 2. a) 2, b) 12.3. (5√2)2=52×(√2)2=25×2=50Cube roots and higher rootsThe cube root of a number, is the number which when cubed gives the original number. Forexample, because 43= 64 we know that the cube root of 64 is 4, written3√64 = 4. All numbers,both positive and negative, possess a single cube root.Higher roots are defined in a similar way: because 25= 32, the fifth root of 32 is 2, written5√32=2.Exercises1. Without usin ga calculator find a)3√27,b)3√125.Answers1. a) 3,b) 5.SurdsExpressions involvin groots, for example√2 and 53√2 are also known assurds. It is usually quiteacceptable to leave an answer in surd form rather than calculatin gits decimal approximation witha calculator.It is often possible to write surds in equivalent forms. For example,√48 can be written as√3×16,that is√3×√16=4√3.Exercises1. Write the followin gin their simplest surd form: a)√180,b)√63.2. By multiplyin gnumerator and denominator by√2 + 1, show that1√2−1is equivalent to√2+1Answers1. a) 6√5, b) 3√7.businesswww.mathcentre.ac.ukcmathcentreMay29,2003