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## Basic Math Facts – Exponents

The recommendations consist of basic material-mathematical-facts-basic advice. Notations allow us to raise numbers, variables, and even expressions to powers, thereby achieving repeated multiplication. In all kinds of mathematical problems, the phenomenon always requires the student to be fully aware of its properties and characteristics. Here we look at the laws, the knowledge of which will allow any student to master this subject.

In the expression 3^2, read “3 squared,” or “3 to the second power,” there is 3. *base* and 2 is the power or exponent. The notation tells us how many times to use the base as a factor. The same goes for variables and variable expressions. In x^3, this means x*x*x. In (x + 1) ^ 2, this means (x + 1) * (x + 1). Symbols are ubiquitous in algebra and indeed in all mathematics, and understanding their properties and how to work with them is very important. Teaching presentations requires students to be familiar with some basic laws and features.

**Product law**

When multiplying expressions involving the same base with different or equal powers, simply write the base with the combination of powers. For example, (x^3)(x^2) is the same as x^(3 + 2) = x^5. To see why this is the case, think of accentuation as pearls on a string. At x^3 = x*x*x, you have three x’s on the wire. At x^2, you have two pearls. So you have five pearls in your product, or x^5.

**Law of Quotients**

When dividing expressions that involve the same base, you simply subtract powers. Thus in (x^4)/(x^2) = x^(4-2) = x^2. Why this is so depends on this* cancellation property* of real numbers. This property states that when the same number or variable appears in both the numerator and the denominator of a fraction, then the term can be canceled. Let’s look at a numerical example to make this completely clear. Make (5*4)/4. Since the 4 appears at the top and bottom of this phrase, we can kill it—well don’t kill, we don’t want to be violent, but you know what I mean—get the 5. Now let’s multiply and divide to see if this agrees with our answer: (5*4)/4 = 20/4 = 5. Check. Thus it holds the property of cancellation. In an expression like (y^5)/(y^3), if we expand it is (y*y*y*y*y)/(y*y*y). Since we have 3 y’s in the denominator, we can use them to cancel the 3 y’s in the numerator to get y^2. This agrees with y^(5-3) = y^2.

**Power of a Law Power**

In an expression like (x^4)^3, we have what is known as a *strength to strength*. A power law states that we simplify by multiplying powers together. Thus (x^4)^3 = x^(4*3) = x^12. If you’re wondering why that is, note that the base of this expression is x^4. The number 3 tells us to use this base 3 times. In this way we will get (x^4)*(x^4)*(x^4). Now we see this as a product of the same base to the same power and thus can use our first property to get x^(4 + 4+ 4) = x^12.

**Property of the Distributor**

This feature tells us how to simplify an expression like (x^3*y^2)^3. To simplify this, we spread the power of 3 inside the parentheses, adding each power to get x^(3*3)*y^(2*3) = x^9*y^6. To understand why this is so, note that the base in the original expression is x^3*y^2. The 3 outer parentheses tells us to multiply this base by itself 3 times. Once you do that and then re-use the expression using both the conjunction and variable properties of the multiplication, then you can apply the first property to get the answer.

**Zero Exponent property**

Any number or variable—except 0—is always 1 to the power of 0. Thus 2^0 = 1; x^0 = 1; (x + 1) ^ 0 = 1. To see why, let’s look at the expression (x^3)/(x^3). This is obviously equal to 1, because any number (except 0) or expression on its own gives this result. Using our equality property, we see that this is x^(3 – 3) = x^0. Since both expressions must give the same result, we get x^0 = 1.

**Negative Exponent Property**

When we subtract a number or variable to a negative number, we end up with *vs*. So 3^(-2) = 1/(3^2). To see why this is so, let’s look at the expression (3^2)/(3^4). If we expand this, we get (3*3)/(3*3*3*3). Using the cancellation property, we end up with 1/(3*3) = 1/(3^2). Using the multiplication property we get (3^2)/(3^4) = 3^(2 – 4) = 3^(-2). Since these two expressions must be equal, we have 3^(-2) = 1/(3^2).

Understanding these six properties of exponents will give students the solid foundation they need to tackle all kinds of pre-algebra, algebra, and even calculus problems. Often times, a student’s feet can be removed with a bulldozer of basic concepts. Study these features and learn them. Then you will be on your way to mathematical mastery.

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