**Week 2: Exponents – Day 5**

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

Another way to phrase this is 2^{50} is equal to what to the 100?

2^{50} = (2^{1/2})^{50×2} = (2^{1/2})^{100}

Therefore a = 2^{1/2} or the square root of 2.

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

By the addition rule of exponents, answer is 3^{1/2 + 1/4}

which is 3^{3/4}

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

If a = 2^{1/2} and b = 2^{1/3} then a^{2} = 2 and b^{3} = 2.

a > 1 because a^{2} > 1^{2}

therefore a^{3} = a^{2}*a = 2a > 2 = b^{3}

since a^{3} > b^{3}, a > b.

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

2^{-2} = 1/4

As we pointed out in problem 3, 2^{1/2} > 1.

Since 1/4 < 1, the bigger number is 2^{1/2}.

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

(1/2)^{-2} = (2/1)^{2} = 4 (remember that the minus sign in the exponent acts as an exchanger of numerators and denominators).

(1/2)^{1/2} = 1^{1/2}/2^{1/2} = reciprocal of 2^{1/2}

As we pointed out in problem 3, the square root of 2 is greater than one, so its reciprocal is less than one.

The bigger number is (1/2)^{-2}.

**Week 2: Exponents – Day 4**

**Week 2: Exponents – Day 3**

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}.

The negative exponent puts 2^{4} in the denominator.

The expression is equivalent to:

^{4}/_{24}

Which is 4/16, which is 1/4.

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

-2^{3} = -(2 x 2 x 2) = -8

2^{-3} = ^{1}/_{23} = 1/8

These are not the same number.

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

(-3)^{-2} = ^{1}/_{(-3)2}

Which is the same as ^{1}/_{(-3)*(-3)}

Which is the same as ^{1}/_{(-3)} x ^{1}/_{(-3)}

All of which are defined and equal to ^{1}/_{9}

On the other hand: -3^{-2} = ^{1}/_{-32}

= ^{1}/_{-9} = – ^{1}/_{9}

So they are not the same number.

4. Simplify 1 – 2^{-1}

This is the same as 1 – 1/2 = 1/2.

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

^{1}/_{2-3} is the reciprocal of 2^{-3}.

2^{-3} = ^{1}/_{23} = ^{1}/_{8}

The reciprocal of ^{1}/_{8} is 8.

Put more simply, a negative exponent flips numerators and denominators. So our original expression can be flipped immediately to 2^{3} which is 8.

**Week 2: Exponents – Day 2**

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?

Saturday is five days after Monday. The bacteria population has multiplied by two five times.

The population is 10 x 2^{5} = 10 x 32 = 320.

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.)

The number of bacteria on Tuesday is 10 x 2 = 20. The number of bacteria on Monday is 10.

The average population growth rate from Monday to Tuesday is (20 – 10) / 1 = 10 bacteria/day.

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

The number of bacteria on Saturday is 320 (see problem 1). The number of bacteria on Friday is 10 x 2^{4} = 10 x 16 = 160.

The average population growth rate from Friday to Saturday is (320 – 160)/1 = 160 bacteria/day.

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?)

The growth rate from Friday to Saturday is much higher than the growth rate from Monday to Tuesday. I might expect the growth rate from Tuesday to Friday to be somewhere in between, if there is a pattern of increasing growth rate.

The number of bacteria on Friday is 160. The number on Tuesday is 20. The number of days between Friday and Tuesday is 3.

Therefore the average population growth rate from Tuesday to Friday is (160 – 20)/3 = 140/3 which is approximately 47 bacteria/day (46 and 2/3 bacteria per day). This is as expected in between the two other average growth rates.

Think about why it is not accurate to say that Population = Rate x Time.

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?

In thirty days, the population is 10 x 2^{30} = 10 x 2^{10} x 2^{10} x 2^{10} which is about 10 x 1000 x 1000 x 1000, approximately ten billion. The petri dish is getting quite crowded.

Week 2: Exponents – Day 1

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

100 + n = 300

Therefore, n = 200.

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

8 = 2^{3}

10 – n = 3

Therefore, n = 7.

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

2n + 3 = 11

2n = 8

Therefore, n = 4.

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

4 = 2^{2}

4^{n} = 2^{2n}

20 = 2n

Therefore, n = 10.

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

9 = 3^{2}

9^{13} = 3^{2×13}

Therefore, n = 26.