Instructional Material on Exponents and Roots 1-26 Exponents described An exponent tells how many times a number is to be used as a factor. For example, in the expression 43, the exponent 3 means that we are to multiply 4 3 4 3 4. The expression 54 means 5 3 5 3 5 3 5. If the exponent is zero, the value of the expression is always 1. Thus, 30 5 100 5 1. If the exponent is 1, the value of the expression is the number whose exponent is 1. Thus, 31 5 3, and 71 5 7. Problems with exponents can be done by remembering that exponents just count factors. For example: Evaluate 34 3 33. Since 34 means use 3 as a factor four times, and 33 means use 3 as a factor three times, and since we then multiply the results, we are using 3 as a factor 3 1 4 or 7 times. Therefore 34 3 33 5 37. Similarly, 45 4 42 5 43, because we are dividing out two of the five factors of 4, leaving three factors of 4, or 43. A negative exponent indicates the reciprocal of a positive exponent. Thus, 1 522 5 2 . Note that 52 3 522 5 50 5 1. 5 SUMMARY I. If two numbers have the same base (e.g., 32, 35 or 72, 76) when we multiply, we keep the base and add the exponents: Examples: 32 3 33 5 35 221 3 222 5 223 32 3 322 5 30 5 1 25 3 223 5 22 76 3 714 5 720 II. If two numbers have the same base (e.g., 32, 35 or 72, 76) when we divide, we keep the same base and subtract the exponents:
100 • gruber’s complete sat math workbook
Examples: 32 4 31 5 31
75
75 4 76 5 721
221 4 222 5 21
68 4 623 5 611 622 4 62 5 624 III. Examples of products of exponents:
4
73
5
72
75 5 71 5 7 74 89 5 86 3 8 87 5 80 5 1 7 8
(32)3 5 32 3 32 3 32 5 3233 5 36 (42)4 5 48 (5 42 3 42 3 42 3 42) When you have a number such as 32 and raise that number to a power. [e.g., (32)3], you multiply exponents and keep the base: (32)3 5 36.
IV. When multiplying two different bases with the same powers, multiply the bases but keep the power. Examples: 24 3 34 5 (2 3 3)4 5 64 35 3 45 5 (3 3 4)5 5 125 71 3 61 5 (7 3 6)1 5 421 5 42 When dividing two different bases with the same powers, divide the bases but keep 4 the power. 2 4 4 Examples: 2 4 3 5 3 5
75 7 5 85 8
67 6 5 5 27 7 3 3
7
Practice Exercises for 1-26 1. Write the product of 23 and 27 as a power of 2. 2. 3. 4. 5. 6.
47 4 42 5 ? 25 5 ? 223 5 ? 32l 3 23 3 40 5 ? 721 3 73 4 75 5 ?
47 7. 4 5 ? 4 32 8. 5 5 ? 3 9. 721 3 722 4 723 5 ?
Diagnostic tests and instructional material • 101
10. (62)3 5 ? (Write as power of 6) 11. (422)23 5 ? (Write as power of 4) 6 12. 5 5 ? (Write as power of 6) 6 13. 6 3 64 3 623 5 ? 14. 221 3 422 3 35 5 ? 15. 12100 5 ? 16. Evaluate: 84 4 44 17. Evaluate: 23 3 33 3 18. Evaluate: 3 93 75 19. What single number raised to a power is 5 ? 4 20. What single number raised to a power is 6215 3 215?
1-27 Square root of a number described The square of a number is the number multiplied by itself. For example, 42 5 4 3 4 5 16. The square root of a given number is a number whose square is the original number. Thus, the square root of 16, written 16 , is 4 since 4 3 4 5 16. (The symbol always means a positive number.) Note: The square root of any number, times itself, is the number. Examples: 4 3 4 5 4; 16 3 16 5 16 The key relationship in simplifying square roots in the square root of a product is the product of the square roots. To simplify a square root, try to find a factor of the number under the square root sign that is a perfect square (4, 9, 16, 25, etc.). If such a factor can be found, then we can simplify.
For example:
32 5 16 3 2 5 16 3 2 5 4 2 ; this cannot be simplified further.
Similarly the sum 3 1 2 cannot be simplified. We can add expressions with square roots only if the numbers inside the square root sign are the same.
50 1 2 . For example: Add Then, 50 1 2 5 5 2 1 2 5 6 2 . Note: Just as 25 3 2 5 50 ,
First,
50 5 25 3 2 5 25 3 2 5 5 2 .
25 25 5 2 2 That is, when we divide square roots, we can divide the numbers inside the square roots and then take the square root of the result.
Division Examples:
15 5 5 3
102 • gruber’s complete sat math workbook
25 5 5 5
36 5 6 6 Multiplication Examples:
20 3 3 5 60 4 3 6 5 24 16 3 2 5 32
Similarly, the cube of a number is the number multiplied by itself three times. For example, since 8 5 (2)(2)(2), 8 is the cube of 2. A cube root of a given number is a number whose cube is the given number. Thus, 2 is a cube root of 8, written 3 8 5 2 . 1
8 can be written as 8 1 3 . Similarly 2 can be written as 2 2 , and so forth. Rationalizing the denominator: Sometimes we wish to have the square root of a number in the numerator instead of in the denominator. Note:
3
Example: Convert the fraction 1 so that the denominator does not contain a 2 square root. Solution: Multiply both denominator and numerator by 2 : 1 2 2 2 3 5 5 . 2 2 2 23 2 Notice that by multiplying the denominator by denominator.
2 , we “rationalize” the
Practice Exercises for 1-27 1. What is the value of the square root of 36? 9. Simplify: 5 1 80 1 25 2. Simplify: 700 10. What is 32 3 4 3. Add:
50 1 18 1 32
4. What is the value of 5. Evaluate: 6. Simplify:
53
10 2
3
64
12. Evaluate:
13. Simplify:
8 1 2 1 16 1 32
7. Evaluate: 64½
11. What is
14. What is 15. If
27 3 51
20 2
45
70 31
49 1 64
27 2 1
6
3 2 14
81
2 is approximately equal to 3 1 8. Convert the fraction so it has a 1.4142, approximate 2 2 simplified denominator. (Hint: Rationalize denominator so denominator does not have 2 in it.)
114 • gruber’s complete sat math workbook
1-26 1. 210 2.
45
3. 32 4.
1 8
1-27 1. 6
8 2 or 2 3 3 1 23 6. 7 or 3 7 5.
7. 43 or 64 23 8. 3 or
1 27
9. 1 10.
66
11. 46 24 12. 6 or
1 64
13. 62 or 36
17. 63 or 216
243 19 or 7 14. 32 32
1 1 18. or 27 3
15. 1
19. (1.75)5
16. 24 or 16
20. 12415
10. 8 2
13. 4 2 2
3
4. 4
7. 8
2. 10 7
5. 5
8.
3 2 2
11. 3
14. 18
3. 12 2
6. 7 2 1 4
9. 5 5 1 5
12. 0
15. 1.4142 or .7071 2
1-28 1. 2. 3. 4. 5.
25 30 (B), (C), (D), (F) 11 3 3 3 2 3 2 Because any even number is divisible by 2, an even number (except 2 since 2 is prime) cannot be prime.
1-29 1. (B) 345 2. (A) 212 and (B) 724 3. (B) 325 and (C) 710 4. (B) 3123 and (C) 2178 5. (A) 20896
6. 11 7. Because it can only be exactly divisible by itself and 1. 8. 2 1 4 1 6 is greater than 1 1 3 1 5. 9. 11 10. 7 3 3 3 3