1. If we know that a^{100} = 2^{50} , what is a?

2. What is 3^{1/2} x 3^{1/4} ?

3. Which is bigger, 2^{1/2} or 2^{1/3} ?

4. Which is bigger, 2^{-2} or 2^{1/2} ?

5. Which is bigger, (1/2)^{-2} or (1/2)^{1/2} ?

1. If we know that a^{100} = 2^{50} , what is a?

2. What is 3^{1/2} x 3^{1/4} ?

3. Which is bigger, 2^{1/2} or 2^{1/3} ?

4. Which is bigger, 2^{-2} or 2^{1/2} ?

5. Which is bigger, (1/2)^{-2} or (1/2)^{1/2} ?

Negative Exponents, Parentheses, Signs and Simplification

“Simplifying” expressions with exponents usually means canceling out all common factors and rewriting them with only positive exponents.

1. Simplify 4 x 2^{-4}.

2. Simplify -2^{3} and 2^{-3}. Are they the same number?

3. Simplify (-3)^{-2} and -3^{-2}. Are they the same number?

4. Simplify 1 – 2^{-1}

5. Simplify ^{1}/_{2-3}

Give exponential answers to these questions – in other words, your answer can involve numbers raised to an exponent.

1. On Monday, 10 bacteria live in a petri dish. Each bacterium divides into two bacteria every twenty-four hours. How many bacteria are in the dish the following Saturday?

2. What is the average rate of bacteria population growth in problem 1 between Monday and Tuesday? (The average growth rate, in bacteria per day, is the number of bacteria on Tuesday minus the number of bacteria on Monday divided by the number of days between Tuesday and Monday.)

3. What is the average rate of bacteria population growth in problem 1 between Friday and Saturday?

4. What is the average rate of bacterial population growth in problem 1 between Tuesday and Friday? (Think about what answer you expect to this problem. Do you expect the same answer as problem 2 or problem 3, or something different?)

5. How many bacteria are in the petri dish one month later (thirty days later)?

Bonus question: since 2^{10} is approximately 1000, approximately how many bacteria are living in the dish after one month – a thousand? million? billion? trillion?

1. If we know that 2^{100} x 2^{n} = 2^{300 }what is n?

2. If we know that 2^{10} / 2^{n} = 8, what is n?

3. If we know that 2^{2n} x 2^{3} = 2^{11} what is n?

4. If we know that 2^{20} = 4^{n} what is n?

5. If we know that 9^{13} = 3^{n} what is n?