Practice Exercises

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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it!

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108 Decimal Fractions

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1 Draw a number line in power-of-10 style that shows the rational numbers from 10 to 100,000 How many orders of magnitude is this 2 Draw a number line in power-of-10 style that shows the rational numbers from 30 to 300,000 How many orders of magnitude is this 3 How many orders of magnitude larger than 330 is 75,000,000 Here s a hint: Express the answer by saying 75,000,000 is between n and n + 1 orders of magnitude larger than 330, where n is a whole number 4 Write the following decimal expressions as combinations of integers and fractions Reduce the fractional part to lowest terms (a) 47 (b) 835 (c) 002 (d) 029 5 Express the numbers from the solutions to Prob 4 as ratios of integers, with the denominator always positive 6 Write the following ratios as decimal expressions (a) 44/16 (b) 81/27 (c) 51/13 (d) 45/800 7 Convert the fraction 1/17 to another fraction whose denominator is a string of 9s 8 Convert the expression 2892892892 to a ratio of integers 9 Suppose that somebody tells us there are two integers so large that it would take a person billions of years to write either of them out by hand We are also told that both of these integers are prime numbers, and that the decimal expansion of their ratio (that is, one of these primes divided by the other) is an endless sequence of digits Is there a repeating pattern to the digits in the decimal expansion 10 Imagine that we come across a gigantic string of digits miles long if we try to write it out and we can t see any pattern We use a computer to examine this number to a thousand decimal places, then a million, then a billion, and we still can t find a pattern Can we ever know for sure whether or not a pattern actually exists, so we can decide whether or not the number is rational Here s a hint: This exercise is meant to force your imagination into overdrive!

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CHAPTER

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Powers and Roots

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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 repeated multiplication When you take a number to an integer root, it s like repeated division But powers and roots go deeper than that! With a few exceptions, you can raise anything to a rational-number power and get a meaningful result

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Integer Powers

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The simplest powers, also called exponential operations, involve multiplying a number or quantity by itself a certain number of times The power is written as a superscript after the quantity to be operated on This operation is sometimes called raising to a power

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Positive integer powers If a is any number and p is a positive integer, the expression a p means a to the pth power, which is a multiplied by itself p times More generally, a doesn t have to be a number It can be a variable or a complicated expression containing numbers and variables Here are some examples of quantities raised to positive integer powers:

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42 x4 (k + 4)7 (abc)4 (m /n)12 where n 0 (x 2 2x + 1)5 Note that in the last expression, the quantity raised to the 5th power actually contains a variable raised to a different power