9514 1404 393
Answer:
27. 15 = 1×15 = 3×5
28. 30 = 1×30 = 2×15 = 3×10 = 5×6
29. 18 = 1×18 = 2×9 = 3×6
30. 17 = 1×17
31. base: 30; power: 2; value: -900
32. base: 12; power: 4; value: 20,736
33. base: 11; power: 3; value: 1,331
Step-by-step explanation:
27-30: The factor pairs can be found by looking at the unique subsets of the set of prime factors. The product of the elements of each subset will be a divisor. The "empty" set can be considered to contain 1. When divisors are listed in numerical order, the factor pairs can be found by pairing factors from each end of the list, working toward the center. If the number is a perfect square, there will be an odd number of divisors, and the center number will be used twice
Examples:
18 = 2×3×3 (prime factorization).
Subsets of these factors are {}, {3}, {3}, {3,3}, {2}, {2,3}, {2,3}, {2,3,3}. Then the divisors are 1, 2, 3, 6, 9, 18. The factor pairs are 1×18, 2×9, 3×6.
25 = 5×5 (prime factorization).
Unique subsets are {}, {5}, {5,5}. Divisors are 1, 5, 25. Factor pairs are 1×25, 5×5.
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The purpose of an exponent is to indicate repeated multiplication. That is, just as 2×3 means 2+2+2 (repeated addition), the use of an exponent means repeated multiplication: 2³ means 2×2×2.
This idea can be extended to give several rules of exponents--rules for multiplication, division, powers, and roots. All of these rules derive from the basic idea of repeated multiplication.
The number that is repeatedly multiplied is called the base. The exponent is called the power.
In plain text, the caret (^) symbol is used to signify an exponent. If an exponent is other than a single number or variable, it must be enclosed in parentheses, so that there is no confusion regarding what the exponent consists of. (This is an "order of operations" issue, since exponentiation must happen before any other operation.) In typeset math expressions, an exponent is rendered in superscript text.
31. -30^2 = -900 . . . base is 30, power is 2. The minus sign is not part of the base. Again, this interpretation is based on the "order of operations".
32. 12^4 = 20,736 . . . base is 12, power is 4.
33. 11^3 = 1,331 . . . base is 11, power is 3
