16D Coreq

In the next preview assignment and in the next class, you will need to evaluate and manipulate exponents.

Exponents

The mathematical operation of exponentiation is denoted using superscript notation:

𝑏^𝑥

Some ways that this can be read include “𝑏 raised to the power of 𝑥”or “𝑏 raised to the𝑥power.” The quantity 𝑏 is called the base, and the quantity 𝑥 is called the exponent. When the exponent is a positive integer, the exponent describes how many times to multiply the base by itself. For example:

2⁴=2⋅2⋅2⋅2=16

When the exponent is 2, we say that we are squaring the base. The quantity

𝑏²

can be read as “𝑏 raised to the power of 2” or “𝑏 raised to the second power,” as described above, but it can also be read as “𝑏 squared.” When the exponent is 3, we say that we are cubing the base. The quantity

𝑏³

can be read as “𝑏 raised to the power of 3” or “𝑏 raised to the third power,” as described above, but it can also be read as “𝑏 cubed.”

Question 1

1) For each of the following collections of three numbers, use exponentiation to combine two of the numbers to obtain the third. For example, if you are given the numbers 2, 4, and 16, you could write 24=16.

Part A: 125, 3, 5

Part B: 3, 9, 2

PartC: 3, 8, 2

Part D: 27, 3, 3

Part E: 5, 2, 32

Part F: 2, 51, 2601

Part G: 5764801, 7, 8

Question 2

2) Fill in the first four rows of the following table to show what happens when you raise the number 2 to different powers. For now, just fill in the blanks corresponding to positive exponents. Do not fill in the shaded cells.

Exponent Exponential expressionEquivalent expression using multiplication and divisionEvaluated expression42⋅2⋅2⋅2163210−1−2−3−4

Question 3

3) In the table, start at the expression 21and work your way up. What happens to the evaluated expression as you move up the table?

Question 4

4) In the table, start at the expression 24 and work your way down. What happens to the evaluated expression as you move down the table?

Question 5

5) Now, fill in the “Evaluated expression” column in the table using the pattern you discovered in the previous question.What you have discovered is that when an exponent is negative, it tells us to take the reciprocal of the result we get when we have a positive exponent. In other words,𝑏−𝑥=1𝑏𝑥

Question 6

6) Using this information, fill in the rest of the table.In the next question, we will explore what happens when the exponent is a unit fraction. A unit fraction is a fraction whose numerator is 1 and whose denominator is a positive integer. The fractions 14and 125are examples of unit fractions.

Question 7

7) Evaluate each of the following using a calculator.

Part A: 1251/3

Part B: 26011/2

Part C: 91/2

Part D: 81/3

Part E: 271/3

Question 8

8) Using what you saw in the previous question, what does it mean to raise a number to the 1/2power? What does it mean to raise a number to the 1/3power?

Question 9

9) Based on your answer to the previous question, what do you think it means to raise a number to the 1/10power?So far, we’ve thought about what happens when you raise a number to a certain power. But we can also turn that question around and ask,“To whatpower must we raise one number to get another number?”

Question 10

10) Using what you know about exponents, answer the following questions.

Part A: To what power must we raise 5 to get 125?

Part B: To what power must we raise 9 to get 3?

Part C: To what power must we raise 2 to get 1/4?

Module 16