Algebra Know-It-ALL

The extra spaces on either side of the decimal point are there to make it easy to distinguish the digit string to its left from ...

106 Decimal Fractions Then write down the part of the expression to the right of the decimal point in this form, with the point ...

The whole-number portion, 23, can be multiplied by 9,999, and the result put into the numerator of a fraction. The denominator s ...

108 Decimal Fractions Draw a number line in power-of-10 style that shows the rational numbers from 10 to 100,000. How many orde ...

Now it’s time to review and expand your knowledge of powers and roots. When you take a number to an integer power, it’s like rep ...

110 Powers and Roots The 0th power By convention, anything raised to the 0th power is equal to 1. Anything except 0 itself, that ...

The constraints are imposed, as with the 0th power, to keep the “powerized” quantities from being equal to 0. If you take a nega ...

112 Powers and Roots The value keeps getting larger without limit, doubling every time. The sequence of values is said to diverg ...

The second power is often called the square, so we can say, “5 squared equals 25.” By defini- tion then 25 1/2= 5 We would say, ...

114 Powers and Roots The third power is often called the cube. We can say, “4 cubed equals 64.” Now if we go with the reciprocal ...

56 = 15,625 so 15,6251/6= 5 − 37 =−2,187 so (−2,187)1/7=− 3 (−5)^9 =−1,953,125, so (−1,953,125)1/9=− 5 64 = 1,296 so 1,2961/4= 6 ...

116 Powers and Roots 15,6251/6= 5 1 1/2= 1 You can indicate that you want to use the negative value by placing minus signs like ...

125 to the −1/3 power. You take the 1/3 power of 125, which is 5, and then take the reciprocal of that, which is 1/5. Mathematic ...

118 Powers and Roots Let’s call this the addition-of-exponents (AOE) rule. We can see how it works by trying out an example with ...

The AOE and SOE rules work not only when the exponents are integers, but for any rational numbers. You might call these facts th ...

120 Powers and Roots When exponents multiply Imagine that we have a number a that is not equal to 0. Suppose p and q are integer ...

exponentsp and q be any rational numbers we want. That gives us the powerful, far-reaching generalized multiplication-of-exponen ...

122 Powers and Roots Are you confused? Let’s review the most important points in this chapter. They can be condensed into six st ...

Practice Exercises This is an open-book quiz. You may (and should) refer to the text as you solve these problems. Don’t hurry! Y ...

124 CHAPTER 9 Irrational and Real Numbers We started out with the natural numbers (or naturals), and took their negatives to get ...