Arithmetic/Exponents/Exercise Answers

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Answers to Exponent Exercises[edit]

1. What are 4^3, 3^4, 1^{250}, {250}^1?

Answers:

a. \ 4^3 = 4 \cdot 4 \cdot 4 = 64
b. \ 3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81
c. \ 1^{250} = 1 \cdot 1 \cdot 1 \cdot \ .\,.\,.\,.\  \cdot 1 \cdot 1 \cdot 1 = 1
d. \ {250}^1 = 250


2. Write these numbers as powers of 2: 128, 8, 1024

Answers:

a. \ 128 = 2^7
b. \ 8 = 2^3
c. \ 1024 = 2^{10}


3. What is (2^3)*(2^2)?

Answer:

(2^3) \cdot (2^2) = 2^{3+2} = 2^5 = 32


4. What is (2^6)/(2^2)?

Answer:

(2^6)/(2^2) = 2^{6-2} = 2^4 = 16 \


5. Harder: Why does 3^0=1? (clue: think about {3^2}/{3^2}, for example)

Answer:

{3^2}/{3^2} = 3^{2-2} = 3^0 \
 3^2 = 3 \cdot 3 = 9
 {3^2}/{3^2} = 9/9 = 1 \
From the first and third equations above, we can see that:
 {3^2}/{3^2} = 3^0 = 1 \
The exponent doesn't have to be 2. The exponent can be any real number and the same logic would work. It was mentioned in the first Arithmetic chapter that a number raised to the 0 power equals 1; i. e.,    
 a^0 = 1 \ .