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**Extra resources for Basic Mathematical Skills**

**Sample text**

Complete the following examples. Show the answer as a base and index. 1. 10 12 x 10- 3 2. 3. i0 6 10 3 104 X 10 2 56 57 Indices A. 10-3 = 109 1. 10 12 X 2. 106 10 3 = 10 6 - 3 = 10 3 3. 104 X 102 = 10 6 In the previous section we learned that to divide by a fraction we invert the fraction and multiply, for example, Similarly, in the following example; 1000 100 + 100 = 1 ~O x 10 ~OO = 1 000000 If we change the above example from numbers to terms of bases and indices we arrive at the following: 2 .

43 Indices 103 is short for 1000 103 = 10 x 10 x 10 = 1000 A. If we had to multiply 102 x 103 we could do it this way. 102 Q. X 103 = 10 x 10 x 10 x 10 x 10 = 10x 10x 10x 10x 10 = 100000 = 105 Use this method to multiply 104 x 105 44 Indices A. 104 X 10 5 = (10 x 10 x 10 x 10) x (10 x 10 x 10 x 10 x 10) = 10x 10x 10x 10x 10x 10x 10x 10x 10 = 1 000 000 000 or 10 9 This method can get a bit tedious, as you have no doubt found out. Now, we have worked out that: 10 2 10 4 X X 10 3 = 10 5 10 5 = 10 9 If we also tell you that: 108 x 10 7 = 10 1 5 What has happened to the index figures of the bases being multiplied to give the index of the answer?

To multiply base and index in this short method, we have said that the base in both cases must be the same. In our example we have a base of 10 and a base of 5 so we cannot do it by the short method we have been looking at. Let us work it out the long way to prove our point. 10 2 X 52 = 10 x 10 x 5 x 5 = 100 x 25 = 2509 Our incorrect answer was 15 4 that is, 15 x 15 x 15 x 15 = 50625. So remember, this short method of adding the indices of the bases being multiplied will not work unless both bases are the same.