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- Basic Math Facts – Exponents
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## Basic Math Facts – Exponents

Exponents include juicy basic math data material. Exponents allow us to raise numbers, variables and even expressions to powers, thus achieving repeated multiplications. The exponent always present in all kinds of mathematical problems requires the student to know its characteristics and properties thoroughly. Here we look at the laws, the knowledge of which will allow any student to master this subject.

In the expression 3^2, which is read “3 squared” or “3 to the second power,” 3 is the *base* and 2 is the power or exponent. The exponent tells us how many times the base must be used as a factor. The same applies to variables and variable expressions. In x^3, this means x*x*x. In (x + 1)^2, this means (x + 1)*(x + 1). Exponents are ubiquitous in algebra, and indeed all of mathematics, and understanding their properties and how to work with them is extremely important. Mastery of exponents requires the student to be familiar with some basic laws and properties.

**Product law**

When multiplying expressions involving the same base to different or equal powers, simply write the base to the sum of the powers. For example, (x^3)(x^2) is the same as x^(3 + 2) = x^5. To see why this is so, think of the exponential expression as beads on a string. At x^3 = x*x*x, you have three x’s (beads) on the string. At x^2, you have two beads. So in the product you have five beads, ox^5.

**Law of the quotient**

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

**Power of a power law**

In an expression like (x^4)^3, we have what is known as a *power to power*. The power of a power law states that we simplify by multiplying powers together. So (x^4)^3 = x^(4*3) = x^12. If you think about why this is, notice that the base of this expression is x^4. The exponent 3 tells us to use this base 3 times. So we would get (x^4)*(x^4)*(x^4). Now we see this as a product of the same base by the same power, and so we can use our first property to get x^(4 + 4+ 4) = x^12.

**Distributive property**

This property tells us how to simplify an expression such as (x^3*y^2)^3. To simplify this, we distribute the power of 3 outside the parentheses inside, multiplying each power to get x^(3*3)*y^(2*3) = x^9*y^6. To understand why this is so, notice that the base of the original expression is x^3*y^2. The 3 outer parentheses tell us to multiply this base by itself 3 times. When you do this and then rearrange the expression using the associative and commutative properties of multiplication, you can apply the first property to get the answer.

**Zero Exponent Property**

Any number or variable—except 0—to the power 0 is always 1. Thus 2^0 = 1; x^0 = 1; (x + 1)^0 = 1. To see why this is so, consider the expression (x^3)/(x^3). This is clearly equal to 1, since any number (except 0) or expression about itself produces this result. Using our quotient property, we see that it equals ax^(3 – 3) = x^0. Since both expressions must give the same result, we get that x^0 = 1.

**Property of negative exponent**

When we raise a number or a variable to a negative integer, we end up with it *reciprocal*. That is 3^(-2) = 1/(3^2). To see why this is so, consider the expression (3^2)/(3^4). If we expand it, we get (3*3)/(3*3*3*3). Using the cancellation property, we end up with 1/(3*3) = 1/(3^2). Using the quotient property we have that (3^2)/(3^4) = 3^(2 – 4) = 3^(-2). Since these two expressions must be equal, we have that 3^(-2) = 1/(3^2).

Understanding these six properties of exponents will provide students with the solid foundation they need to tackle all kinds of pre-algebra, algebra, and even calculus problems. Often, a student’s pitfalls can be removed with the excavator of fundamental concepts. Study these properties and learn them. Then you’ll be on your way to math mastery.

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