You are correct. The 3rd power of 4 = 43, not 34.
We have seen that in a multiplication the numbers that are multiplied together are called the factors and the result is called the product. Thus, in the multiplication 2 × 3 = 6, the numbers 2 and 3 are the factors and 6 is the product.
When the same factor is used more than once in a multiplication, the resulting product is said to be a
power of the original factor. Thus if the number 6 is used as a factor twice, we have:
6 × 6 = 36
The number 36 is a power of 6. Specifically, since 6 was used as a factor only twice in our multiplication, 36 is the second power of 6. If 6 were to be used as a factor three times, we should have
6 × 6 × 6 = 216
The number 216 is the third power of 6.
The general expression which means “use a number as a factor a certain number of times is bn. The b stands for the base, or the number which is to be used as a factor. The n stands for the exponent or the number which tells how many times the base is to be used as a factor.
The exponent is written as a small number just above and immediately following the base. Thus if we wish to use a given base, 2, as a factor 4 times, we write 24.
This term, 24 means 2 × 2 × 2 × 2, which is equal to 16.
The term 24is not to be thought of as an “artificial” way of writing the “real” number 16. Any set of symbols we agree to use for a given quantity is just as valid as any other symbol. Thus 24 and 16 and XVI are just three different symbolic expressions denoting the same quantity. Similarly, 106 is a perfectly valid way of expressing the quantity we usually write as 1,000,000.
There are several ways we can talk about using the same number as a factor to obtain a product. The process is sometime called the expansion of a base. When we write 106, we mean “ using 10 as a factor 6 times” or “raising 10 to the 6th power.”
Raisning a number to the second power is often called squaring it. Thus the square of 3 = 32 = 9. Raising a number to the third power may be called cubing it. Thus the cube of 3 = 33 = 27.
The terms “square” and “cube” obviously come from geometry. The area of a square that measures 3 feet on each side can be found by squaring 3 to get 9 square feet. Similarly, the number of cubic feet in a cube that measures 3 feet on each side is 33, or 27 cubic feet.
An exponent of 1 means “raise the base to the first power”. The first power of any number is always the same as the number itself. b1 = b, 21 = 2 and 41 = 4.
Understanding factors, bases and exponents is of considerable importance in both practical and theoretical mathematics and in all science. In physics, chemistry and astronomy, for example, many quantities are routinely expressed as factors and as powers, because they are so large or so small as to be virtually unmanageable otherwise,
The entire system of logarithms, which simplifies calculations with numbers containing many digits, is based upon the principle of raising a given base to various powers. So knowledge of bases and powers is essential to any real understanding of logarithms, or of any of the computational devices based on logarithmic principles, such as the slide rule.
We are taking up powers and factors here as a means of understanding the number systems used in digital computers. But the principles discussed here will be useful to you in many other areas.
In order to fix firmly in your mind what you have learned about factors and powers of numbers, you must practice using your new knowledge by solving problems. Self-Test Questions are provided for this purpose. When you have finished reading this material, you should turn to the Self-Test Questions on Lesson 1.
The answer to each Self-Test Question appears at the bottom of the page in the text which discusses the information needed to answer the question correctly. When you have determined your answer to a Self-Test Question, you can check your answer by referring to the page number indicated after the question. If your answer is wrong, read the entire page. If your answer is right, return immediately to the Self-Test Question page and answer the next question.
The Self-Test Questions for Lesson 1 are in the Appendix. Turn to them now.
