Algebra For Beginners - insight the Law of Exponents

Adding Exponents Worksheet - Algebra For Beginners - insight the Law of Exponents

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In this installment of Algebra for Beginners, we will look at a topic that often causes students to be unsuccessful--exponents. As is true with many Algebra topics, students tend to bring a good insight of exponents from arithmetic, but then have trouble transferring that skill from numbers to the variables used in Algebra. Hopefully, we can literal, this problem.

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Adding Exponents Worksheet

There are two former 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 precisely cut off rules for handling the different types of exponents. This means that, unfortunately, there is Not a particular Law of Exponents.

The purpose of this narrative is to two-fold: (1) clear up the coefficient versus exponent confusion, and (2) discuss the One property that most population reconsider "the law of exponents." The other situations consuming exponents will be discussed in other articles.

Coefficients versus Exponents:

We need to begin with a relate of prominent terminology. Remember that Algebraic terms are combinations of numbers and/or variables using multiplication/division--Not addition/subtraction. For example: x, 5y, 7, a/b, 2a^3 are all algebraic terms. The number in front, even if that number is an understood one, is called the coefficient of the term, while the raised numbers on the variables are called exponents. Again, if those exponents are not visible they are understood to be ones.

Both coefficients and exponents riposte a "how many" question. The coefficient tells us how many times the variable part of the term was or could be added together. Thus, 4x = x + x + x + x. The term 4x means that x was Added to itself 4 times. An exponent tells us how many times its variable was or could be written as multiplication. In the term 4x^2, the x^2 means (x)(x), so 4x^2 = 4(x)(x).

The Law of Exponents:

The multiplication interpretation for exponents seems very straightforward. It is logical that 4^3 means (4)(4)(4). Right? Remember, though, that exponents can be any kind of number, not just obvious integers. As you look at terms like 4^(-1) or 4^(1/2) or 4^pi or even 4^0, you perceive that multiplication doesn't seem to apply. This is the reckon that there is precisely no particular rule for all of these cases. These "unusual" situations will be discussed in other articles.

What most population think of as The Law of Exponents deals with two different situations consuming integer exponents. The first situation looks like (x^2)(x^3)(x^2)(x). The second situation looks like (x^2)^3. Both of these situations obviously can be simplified, but this is where students get stumped. One formula calls for addition of exponents and the other calls for multiplication of the exponents, but which is which?

The key to simplifying these expressions consuming exponents is to fall back on the definition of exponents. For example, the first situation, (x^2)(x^3)(x^2)(x) should be viewed as multiplying like bases. To simplify this expression, we use the exponent definition to develop the expression as (x x)(x x x)(x x)(x) which shows x multiplied by itself 8 times or x^8. Consideration 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 raising a power to someone else power. Again, we can simplify by relying on the definition of exponents. (x^2)^3 = (x x)(x x)(x x) = x^6. Consideration 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 reconsider there to be a particular Law of Exponents, it would look like the following:

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 a power, keep the base and multiply the exponents. Example: (b^3)^5 = b^(3*5) = b^15.

To become successful in working with exponents quickly, you need to memorize the above rules in both words and symbols and you need to practice, practice, practice! But, you do not precisely need to memorize these rules as long as you understand the definition of exponents. The rules above are Shortcuts, but falling back on the definition will all the time get you to the literal, simplification. If you have trouble remembering when to add exponents and when to multiply exponents, then just rewrite the expressions using the exponent definition and the ensue will show itself. Isn't math Great?!

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