Arithmetic of Computers

Arithmetic of Computers

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Lesson 2

The Arithmetic of Powers

Multiplication by addition, and division by subtraction

Page 63

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Your answer :
(b3)(b4) = b7.
You are correct. The rule is: when two numbers that are powers of the same base are multiplied, the product is a power of that same base, and the exponent of the product is the sum of the exponents of the two numbers that were multiplied. This rule is summed up as
(bm)(bn) = b(m + n)
We can show that this is reasonable by considering that the total number of times the base, b, is used as a factor in the product will be simply the total number of times it is used as a factor in the two numbers that were muliplied. This in the multiplication
(32)(33)  = 35,  32 × 33  = 35,  (3 × 3) × (3 × 3 × 3)  = (3 × 3 × 3 × 3 × 3).
Very well, a little practice will not be amiss. One of the sets of multiplications shown below contains a mistake. Which is it?
Answer :
(52)(54) = 56     (12)(13) = 15 = 1 82 = 64     (32)(32) = 81 ⎫⎬⎭ This set contains a mistake.

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103 = 1000     (bm)(b4) = b(m + 4) (42)(41) = 64     (84)(86) = 810 ⎫⎬⎭ This set contains a mistake.

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(23)(22) = 32     (bm)(b4) = b(m + 4) (9n)(9m) = 9(n + m)     (84)(82) = 88 ⎫⎬⎭ This set contains a mistake.

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(b3)(b4) = b7     (102)(102) = 10, 000 (52)(51) = 125     (122)(12) = 123 ⎫⎬⎭ This set contains a mistake.

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Answer to Self-Test Question 8, Lesson 2 :
(bm)(bn) = b(m + n).

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