2. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Working with Fractions Continued 3 4, Proper fractions Proper fractions consist of the numerator that is always smaller than the denominator. For example: 7 8, 13 32, 1 2 Etc. 12 8, Improper fractions Improper fractions are just the opposite. The numerator that is always larger than the denominator. For example: 5 2, 44 18, 12 3 Etc. Mixed number fractions Mixed number fractions contain both – whole numbers and a proper fractions. All improper fractions could, and in many cases, should be converted to mixed number fractions. For example: 3 4, 2 5 8, 3 7 16 Etc. 7 In real world applications, the mixed number fractions are used quite often. For instance, when someone sais “one and a have pounds”, they are using a mixed number fraction. Mathematically they automatically assume the following number 1 2 1 lbs.

3. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Converting improper fractions to mixed number fractions 25 11 Let us take this improper fraction: To convert this to the mixed number fraction, we would have to divide the numerator by the denominator. In other words, we have to find Out how many times 11 fits into 25. In this case it’s quite simple 11 can only go into 25 two times (this will be our whole number later). So, 11 x 2 = 22. Now we simply subtract 22 from 25. That of course gives us 3. 3 is the remainder that’s left from the subtraction. Therefore our entire new number will be: 3 11 2 Additional example: 45 4 First divide 4 into 45 45 ÷ 4 = 11 Although when you use the calculator the actual answer is 11.25 we would just need the whole number 11 and ignore the decimal. So 11 would be our whole number of the mixed fraction. Now we multiply 11 by 4. That gives us 44. Next, 45 – 44 = 1 That’s our remainder. Therefore our entire new number will be: 1 4 11

4. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Converting mixed number fractions into improper fractions: 45 4 Sometimes, for the purpose of easier calculations, the mixed number fractions have to be converted to improper fractions. The procedure is quite simple. Let’s use the mixed fraction from the last example and convert it to the improper fraction: 1 4 11 The easiest way to do that is simply multiply 11 and 4, add the numerator, and put the result as the numerator, on top of our fraction. That’s all you have to do. Let’s see, 11 x 4 = 44 44+1 = 45 Therefore the final result is: Let us try two other examples: 7 8 5 8 x 5 = 40 40 + 7 = 47 The final answer is 47 8 5 16 4 16 x 4 = 64 64 + 5 = 69 The final answer is 69 16

5. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Advanced Exponents Topics: As mentioned in the earlier sections, exponents or indicators of “power”, simply imply how many times the number or a variable is multiplied by itself. Here are some examples: 2 4 = 2 x 2 x 2 x 2 = 16 5 3 = 5 x 5 x 5 = 125 8 2 = 8 x 8 = 64 a 4 = a x a x a x a 3ab 3 = 3ab x 3ab x 3ab It is slightly different when you have negative numbers or variables. For example: -3 4 = -(3 x 3 x 3 x 3) = - 81 -12 2 = -(12 x 12) = - 144 -8 3 = -(8 x 8 x 8) = - 512 So as you can see, you can simply ignore the minus sign, do a regular exponent operation (multiply, of course) and then put the minus sign in front of the result. So that the above examples will look like this: -3 4 = - 81 -12 2 = - 144 -8 3 = - 512

6. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Some additional rules about exponents: Any number or variable to the power of 1 equals the same number or variable for example: 4 1 = 4 12 1 = 12 x 1 = x (2xy) 1 = 2xy Any number or variable to the power of 0 equals 1 for example: 4 0 = 1 12 0 = 1 x 0 = 1 (2xy) 0 = 1 Multiplying numbers with exponents: When multiplying numbers or variables with exponents that have the same base value, simply add the power values, but leave the number unchanged. For example: 4 3 · 2 4 = 8 (3+4) or 8 7 12 5 · 12 4 = 12 9 x 3 · x 8 = x 11 2xy 2 · 2xy 4 = 2xy 6 40 7 · 40 4 = 40 11

7. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Dividing numbers or variables with exponents: When dividing numbers or variables with exponents that have the same base value, simply subtract the power values and also leave the number unchanged. For example: 3737 3232 = 3 (7-2) = 3 5 x4x4 x2x2 = x (4-2) = x 2 2a 7 ÷ 2a 3 = 2a 4 x 5 ÷ x 1 = x 4 this is the same as x 5 ÷ x = x 4 (power of 1 is always omitted) Multiplying or dividing the results within the parenthesis raised to exponents : When you multiply or divide the terms within the parenthesis and raised to a power, you have to raise all terms to the same power individually. For example: (3 x 5) 2 = 3 2 x 5 2 (24 ÷ 6) 3 = 24 3 ÷ 6 3 (a x b) 4 = a 4 x b 4

8. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Number raised to two exponents : When the number or variable is raised to two exponents, you can simply multiply the exponents together. For example: (3 3 ) 2 = 3 6 (12 2 ) 5 = 12 10 (a 4 ) 2 = a 8 (3xy 3 ) 4 = 3x 4 y 12 Please note in the fourth example x is also raised to the fourth power because x is also within the parenthesis and needs to be raised to the appropriate power shown on the outside. This cannot be avoided and always needs to be addressed. Negative exponents: When working with negative exponents, for easier calculations, you can convert the negative exponent into a positive exponent by simply creating a reciprocal of the number or a variable. Here’s how it’s done: 14 4 -4 = 1 12 3 12 -3 = } ¼ is a reciprocal of 4 1 x5x5 x -5 = 1 X -4 x4x4 = X4X4 1 = Here are 3 additional examples:

9. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI Working with radicals Radicals are special mathematical technique allowing you to find a “root” or a certain number or variable. You may even say that a root is the opposite of an exponent. For example: a square root of number 49 is 7. But number 7 raised to the second power (squared) is 49. On the next page you see a radical symbol together with some associated terminology: 144 2 Radical (or “root” symbol) Index (or power level) Radicand (or base number) 144 2 Typically, an index of 2 (meaning a SQUARE ROOT) is not shown. A regular root symbol automatically assumes a number 2. Any other index power number will have to be shown. 144 =

10. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI From here on, all radical symbols in this course will be called a root Here are some examples of numbers and variables with roots: 4964 3 (a+b) 4 Here are some easy to recognize square roots (remember, square means to the power of 2): 1 = 1 4 = 2 9 = 3 16 = 4 25 = 5 36 = 6 49 = 7 64 = 8 81 = 9 100 = 10 Here is the basic principle of how roots and powers work: Let us say that I would raise number 6 to the second power. The answer will be 6 x 6 = 36. But if I wanted to find the square root of 36 it would be 6. 36 = 6 6 2 = 36 Note: You can use your scientific calculator to find roots of any number. You can even use a free calculator in Windows or Mac computers. You just have to set it to the Scientific mode. Just for exercise try to figure these roots using your scientific calculator and see if you get the same numbers.

11. Previous Page Next Page EXIT Created by Professor James A. Sinclair, Ph.D. MMXI = 5 125 3 Working with cubic roots: When the number is raised to the third power, for instance 5 3 or (a+b) 3 it is typically called that the number is “cubed”. It always mean that the number is raised to the third power. Here is an example of cubed terms and their roots: 5 3 = 5 x 5 x 5 = 125 And here is a cubic root of 125 Here are some easy to recognize cubic roots (remember, cubic means to the power of 3): 1 3 = 1 216 3 = 6 8 3 = 2 343 3 = 7 27 3 = 3 512 3 = 8 64 3 = 4 729 3 = 9 125 3 = 5 1,000 3 = 10 Just for exercise try to figure these roots using your scientific calculator and see if you get the same numbers.