Algebra I Workbook For Dummies Page 5
so the improper fraction is .
2. Change the mixed number to an improper fraction. The answer is .
, so the improper fraction is .
3. Change the improper fraction to a mixed number. The answer is .
Think of breaking up the fraction into two pieces: One piece is the whole number 3, and the other is the remainder as a fraction, .
4. Change the improper fraction to a mixed number. The answer is .
The negative part of the fraction comes in at the beginning and at the end.
Do the division, using just the positive fraction:
5. Find an equivalent fraction with a denominator of 28 for . The answer is . To get 28 in the denominator, multiply 7 by 4:
6. Find an equivalent fraction with a denominator of 30 for . The answer is .
7. Reduce this fraction: . The answer is . 4 is the greatest common divisor of 16 and 60 because and . So multiply by . You get
8. Reduce this fraction: . The answer is . 21 is the largest common divisor of 63 and 84 because and 84 = 4 × 21. So
9. Solve for x: . The answer is .
j Solve for x: . The answer is .
11. Solve for x: . The answer is .
12. Solve for x: . The answer is .
13. A recipe calls for 2 teaspoons of cinnamon and 4 cups of flour. You need to increase the flour to 6 cups. To keep the ingredients proportional, how many teaspoons of cinnamon should you use? Hint:
Fill in the proportion. and let x represent the new cinnamon. The answer is .
14. A factory produces two faulty iPods for every 500 iPods it produces. How many faulty iPods would you expect to find in a shipment of 1,250? The answer is .
15. Rewrite the fractions and with a common denominator. The answers are and .
The largest common factor of 7 and 8 is 1. So the least common denominator is 56: . Here are the details:
and
16. Rewrite the fractions and with a common denominator. The answers are and . The largest common factor of 12 and 18 is 6. The least common denominator is 36 . Here are the details:
and
17. Rewrite the fractions and with a common denominator. The answers are and . The largest common factor of x and 6 is 1. The least common denominator is their product: 6x. Break it down: and .
18. Rewrite the fractions and with a common denominator. The answers are and . The largest common factor of x and x + 6 is 1. Their least common denominator is their product: x(x + 6). Here’s the long of it: and .
19. Rewrite the fractions , , and with a common denominator. The answers are , , and . The least common denominator of fractions with denominators of 2, 3, and 5 is 30. Write it out:
20. Rewrite the fractions , , and with a common denominator. The answers are , , and . The last two denominators, and , have a common factor of x. And the product of all three denominators is 6x2. Divide the product by x and you get 6x. In long hand:
21. because the least common denominator is 24.
22. because the least common denominator is 15.
Or, leaving the whole number parts separate:
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35. Change to a decimal. The answer is 0.6. Dividing, you have .
36. Change to a decimal. The answer is .
because
37. Change to a decimal. The answer is .
because
38. Change 0.45 to a fraction. The answer is because .
39. Change to a fraction. The answer is .
because two digits repeat .
40. Change to a fraction. The answer is .
because three digits repeat: .
41. . Simplify inside the parentheses first.
42. . By the order of operations, you multiply first and then subtract 3 from the result.
Chapter 4
Exploring Exponents
In This Chapter
Working with positive and negative exponents
Recognizing the power of powers
Operating on exponents
Investigating scientific notation
In the big picture of mathematics, exponents are a fairly new discovery. The principle behind exponents has always been there, but mathematicians had to first agree to use algebraic symbols such as x and y for values, before they could agree to the added shorthand of superscripts to indicate how many times the value was to be used. As a result, instead of writing x · x · x · x · x, you get to write the x with an exponent of 5: x5. In any case, be grateful. Exponents make life a lot easier.
This chapter introduces to you how exponents can be used (and abused), how to recognize scientific notation on a calculator, and how eeeeeasy it is to use e. What’s this “e” business? The letter e was named for the mathematician Leonhard Euler; the Euler number, e, is approximately 2.71828 and is used in business and scientific calculations.
Multiplying and Dividing Exponentials
The number 16 can be written as 24, and the number 64 can be written as 26. When multiplying these two numbers together, you can either write 16 × 64 = 1,024 or you can multiply their two exponential forms together to get 24 × 26 = 210, which is equal to 1,024. The computation is easier — the numbers are smaller — when you use the exponential forms. Exponential forms are also better for writing very large or very small numbers.
To multiply numbers with the same base (b), you add their exponents. The bases must be the same, or this rule doesn’t work.
bm × bn = bm + n
When numbers appear in exponential form, you can divide them simply by subtracting their exponents. As with multiplication, the bases have to be the same in order to perform this operation.
When the bases are the same and two factors are divided, subtract their exponents:
Q. 35 × 3–3 × 7–2 × 7–5 =
A. 32 × 7–7. The two factors with bases of 3 multiply as do the two with bases of 7, but they don’t mix together. The negative exponents probably look intriguing. You can find an explanation of what they’re all about in the “Using Negative Exponents” section later in this chapter.
35 × 3–3 × 7–2 × 7–5 = 35 + (–3) × 7–2 + (–5) = 32 × 7–7.
Q. 56 × 5–7 × 5 =
A. 1. In this case, the multiplier 5 is actually 51 because the exponent 1 usually isn’t shown. Also, 50, or any non-zero number raised to the zero power, is equal to 1.
So 56 × 5–7 × 5 = 56 + (–7) + 1 = 50 = 1.
Q.
A. 3.
.
Q.
A. 41,472. The bases of 8 and 3 are different, so you have to simplify the separate bases before multiplying the results together.
1. 23 × 24 =
Solve It
2. 36 × 3–4 =
Solve It
3. 25 × 215 × 34 × 33 × e4 × e6 =
Solve It
4. 7–3 × 32 × 5 × 79 × 54 =
Solve It
5.
Solve It
6.
Solve It
Raising Powers to Powers
Raising a power to a power means that you take a number in exponential form and raise it to some power. For instance, raising 36 to the fourth power means to multiply the sixth power of 3 by itself four times: 36 × 36 × 36 × 36. As a power of a power, it looks like this: . Raising something to a power tells you how many times it’s multiplied by itself. The rule for performing this operation is simple multiplication.
When raising a power to a power, don’t forget these rules:
. So to raise 36 to the fourth power, write .
and .
and .
These rules say that if you multiply or divide two numbers and are raising the product or quotient to a power, then each factor gets raised to that power. (Remember, a product is the result of multip
lying, and a quotient is the result of dividing.)
Q.
A. 3–28 × 542
Q.
A.
7.
Solve It
8.
Solve It
9.
Solve It
10.
Solve It
11.
Solve It
12.
Solve It
Using Negative Exponents
Negative exponents are very useful in algebra because they allow you to do computations on numbers with the same base without having to deal with pesky fractions.
When you use the negative exponent , you are saying and also .
So if you did problems in the sections earlier in this chapter and didn’t like leaving all those negative exponents in your answers, now you have the option of writing the answers using fractions instead.
Another nice feature of negative exponents is how they affect fractions. Look at this rule:
A quick, easy way of explaining the preceding rule is to just say that a negative exponent flips the fraction and then applies a positive power to the factors.
Q.
A. 2. Move the factors with the negative exponents to the bottom, and their exponents then become positive. Then you can reduce the fraction.
Q.
A. . First flip and then simplify the common bases before raising each factor to the second power.
13. Rewrite , using a negative exponent.
Solve It
14. Rewrite , getting rid of the negative exponent.
Solve It
15. Simplify , leaving no negative exponent.
Solve It
16. Simplify , leaving no negative exponent.
Solve It
Writing Numbers with Scientific Notation
Scientific notation is a standard way of writing in a more compact and useful way numbers that are very small or very large. When a scientist wants to talk about the distance to a star being 45,600,000,000,000,000,000,000,000 light years away, having it written as 4.56 × 1025 makes any comparisons or computations easier.
A number written in scientific notation is the product of a number between 1 and 10 and a power of 10. The power tells how many decimal places the original decimal point was moved in order to make that first number be between 1 and 10. The power is negative when you’re writing a very small number (with four or more zeros after the decimal point) and positive when writing a very large number with lots of zeros in front of the decimal point.
Q. 32,000,000,000 =
A. 3.2 × 1010. Many modern scientific calculators show numbers in scientific notation with the letter E. So if you see 3.2 E 10, it means 3.2 × 1010 or 32,000,000,000.
Q. 0.00000000032 =
A. 3.2 × 10–10 (In a graphing calculator, this looks like 3.2 E –10.)
17. Write 4.03 × 1014 without scientific notation.
Solve It
18. Write 3.71 × 10–13 without scientific notation.
Solve It
19. Write 4,500,000,000,000,000,000 using scientific notation.
Solve It
20. Write 0.0000000000000003267 using scientific notation.
Solve It
Answers to Problems on Discovering Exponents
This section provides the answers (in bold) to the practice problems in this chapter.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13. Rewrite , using a negative exponent. The answer is .
14. Rewrite , getting rid of the negative exponent. The answer is .
15. Simplify , leaving no negative exponent. The answer is .
16. Simplify , leaving no negative exponent. The answer is .
17. Write without scientific notation. The answer is 403,000,000,000,000. Move the decimal point 14 places to the right.
18. Write without scientific notation. The answer is 0.000000000000371. Move the decimal point 13 places to the left.
19. Write 4,500,000,000,000,000,000 using scientific notation. The answer is .
20. Write 0.0000000000000003267 using scientific notation. The answer is 3.267 × 10–16.
Chapter 5
Taming Rampaging Radicals
In This Chapter
Making radical expressions simpler
Trimming down radical fractions
Working in fractional exponents
Performing operations using fractional exponents
The operation of taking a square root, cube root, or any other root is an important one in algebra (as well as in science and other areas of mathematics). The radical symbol () indicates that you want to take a root (what multiplies itself to give you the number or value) of an expression. A more convenient notation, though, is to use a superscript, or power. This superscript, or exponent, is easily incorporated into algebraic work and makes computations easier to perform and results easier to report.
Simplifying Radical Expressions
Simplifying a radical expression means rewriting it as something equivalent using the small-est possible numbers under the radical symbol. If the number under the radical isn’t a perfect square or cube or whichever power for the particular root, then you want to see whether that number has a factor that’s a perfect square or cube (and so on) and factor it out.
The root of a product is equal to the product of two roots containing the factors. The rule is or, more generally, .
In the following two examples, the numbers under the radicals aren’t perfect squares, so the numbers are written as the product of two factors — one of them is a perfect square factor. Apply the rule for roots of products and write the expression in simplified form.
Q.
A. . The number 40 can be written as the product of 2 and 20 or 5 and 8, but none of those numbers are perfect squares. Instead, you use 4 and 10 because 4 is a perfect square.
Q.
A. . In this example, you can’t add the two radicals together the way they are, but after you simplify them, the two terms have the same radical factor, and you can add them together.