In this installment of Algebra for Beginners, we’ll look at a topic that often keeps students from succeeding: exponents. As with many topics in algebra, students tend to have a good understanding of exponents in arithmetic, but then have trouble transferring that skill from numbers to the variables used in algebra. Hopefully, we can correct this problem.
There are two main reasons algebra students get stumped by exponents. The first is that students confuse coefficients and exponents. The second is that because exponents can be any number (integers, fractions, decimals, radicals, positive, negative, and even zero), there are actually separate rules for handling the different types of exponents. This means that, unfortunately, there is NOT a single Law of Exponents.
The purpose of this article is twofold: (1) to clear up the confusion between the coefficient and the exponent, and (2) to discuss the UNIQUE property that most people consider “the law of exponents.” The other situations involving exponents will be discussed in other articles.
Coefficients versus Exponents:
We need to start with a review of important terminology. Remember that algebraic terms they are combinations of numbers and/or variables using multiplication/division—NOT addition/subtraction. For example: x, 5y, 7, a/b, 2a^3 are algebraic terms. The number that is in front, even if that number is understood, is called the coefficient of the term, while the numbers raised over the variables are called exponents. Again, if those exponents are not visible, they are understood to be visible.
Both coefficients and exponents answer a “how many” question. The coefficient tells us how many times the variable part of the term was or could be additional together. So 4x = x + x + x + x. The term 4x means that x ADDS itself 4 times. An exponent tells us how many times its variable was or could be written as multiplication. In the term 4x^2, x^2 means (x)(x), so 4x^2 = 4(x)(x).
The Law of Exponents:
The interpretation of the multiplication of exponents seems very simple. It stands to reason that 4^3 means (4)(4)(4). Right? Remember, though, that exponents can be any type of number, not just positive integers. When you look at terms like 4^(-1) or 4^(1/2) or 4^pi or even 4^0, you realize that multiplication doesn’t seem to apply. This is the reason why there is really no single rule for all these cases. These “unusual” situations will be discussed in other articles.
What most people think of as The Law of Exponents deals with two different situations involving integer exponents. The first situation looks like (x^2)(x^3)(x^2)(x). The second situation looks like (x^2)^3. Obviously, both situations can be simplified, but this is where students get stumped. One method requires the addition of exponents and the other requires the multiplication of exponents, but which is which?
The key to simplifying these expressions involving exponents is refer to the definition of exponents For example, the first situation, (x^2)(x^3)(x^2)(x) should look like multiplying as bases. To simplify this expression, we use the definition of exponent to expand the expression as (xx)(xxx)(xx)(x) which shows x multiplied by itself 8 times or x^8. Notice that the sum of the exponents, 2 + 3 + 2 + 1 = 8, but we didn’t need to know that shortcut to simplify the expression.
The second situation, (x^2)^3 is expressed as raise one power to another power. Again, we can simplify by relying on the definition of exponents. (x^2)^3 = (xx)(xx)(xx) = x^6. Notice that in (x^2)^3, multiplying the exponents produces 6, but again, we didn’t need to know the shortcut to simplify the expression.
Thus, if we are to consider that there is only one Law of Exponents, it would look like this:
Law of exponents:
(a) To multiply like bases, keep the base and add the exponents. Example: (y^5)(y)(y^3) = y^(5 + 1 + 3) = y^9.
(b) To raise a power to another power, keep the base and multiply the exponents. Example: (b^3)^5 = b^(3*5) = b^15.
To be successful in working with exponents. quickly, you need to memorize the above rules both in words and symbols and you need to practice, practice, practice! But, you don’t really need to memorize these rules as long as you understand the definition of exponents. The above rules are DIRECT ACCESS, but resorting to the definition will always get you to the correct simplification. If you’re having trouble remembering when to add exponents and when to multiply exponents, simply rewrite the expressions using the definition of exponent and the result will display itself. Isn’t math great?