Arithmetic/Exponents/Exercise Answers

Answers to Exponent Exercises

1. What are 4³, 3⁴, 1²⁵⁰, 250¹?

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 (2³) * (2²)?

Answer:

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

4. What is (2⁶) / (2²)?

Answer:

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

5. Harder: Why does 3⁰ = 1? (clue: think about 3² / 3², 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 \$ .