Exponents comprise a juicy tidbit of basic-math-facts material. Exponents allow us to raise numbers, variables, and even expressions to powers, thus achieving repeated multiplication. The ever present exponent in all kinds of mathematical problems requires that the student be thoroughly conversant with its features and properties. Here we look at the laws, the knowledge of which, will allow any student to master this topic.
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 to use the base as a factor. The same applies to variables and variable expressions. In x^3, this mean x*x*x. In (x + 1)^2, this means (x + 1)*(x + 1). Exponents are omnipresent in algebra and indeed all of mathematics, and understanding their properties and how to work with them is extremely important. Mastering exponents requires that the student be familiar with some basic laws and properties.
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 pearls on a string. In x^3 = x*x*x, you have three x’s (pearls) on the string. In x^2, you have two pearls. Thus in the product you have five pearls, or x^5.
When dividing expressions involving the same base, you simply subtract the powers. Thus in (x^4)/(x^2) = x^(4-2) = x^2. Why this is so depends on the cancellation property of the real numbers. This property says that when the same number or variable appears in both the numerator and denominator of a fraction, then this term can be canceled. Let us look at a numerical example to make this completely clear. Take (5*4)/4. Since 4 appears in both the top and bottom of this expression, we can kill it—well not kill, we don’t want to get violent, but you know what I mean—to get 5. Now let’s multiply and divide to see if this agrees with our answer: (5*4)/4 = 20/4 = 5. Check. Thus this cancellation property holds. In an expression such as (y^5)/(y^3), this is (y*y*y*y*y)/(y*y*y), if we expand. Since we have 3 y’s in the denominator, we can use those to cancel 3 y’s in the numerator to get y^2. This agrees with y^(5-3) = y^2.
Power of a Power Law
In an expression such as (x^4)^3, we have what is known as a power to a power. The power of a power law states that we simplify by multiplying the powers together. Thus (x^4)^3 = x^(4*3) = x^12. If you think about why this is so, notice that the base in this expression is x^4. The exponent 3 tells us to use this base 3 times. Thus we would obtain (x^4)*(x^4)*(x^4). Now we see this as a product of the same base to the same power and can thus use our first property to get x^(4 + 4+ 4) = x^12.
This property tells us how to simplify an expression such as (x^3*y^2)^3. To simplify this, we distribute the power 3 outside 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 in the original expression is x^3*y^2. The 3 outside parentheses tells us to multiply this base by itself 3 times. When you do that and then rearrange the expression using both the associative and commutative properties of multiplication, you can then apply the first property to get the answer.
Zero Exponent Property
Any number or variable—except 0—to the 0 power is always 1. Thus 2^0 = 1; x^0 = 1; (x + 1)^0 = 1. To see why this is so, let us consider the expression (x^3)/(x^3). This is clearly equal to 1, since any number (except 0) or expression over itself yields this result. Using our quotient property, we see this is equal to x^(3 – 3) = x^0. Since both expressions must yield the same result, we get that x^0 = 1.
Negative Exponent Property
When we raise a number or variable to a negative integer, we end up with the reciprocal. That is 3^(-2) = 1/(3^2). To see why this is so, let us consider the expression (3^2)/(3^4). If we expand this, we obtain (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 that (3^2)/(3^4) = 3^(2 – 4) = 3^(-2). Since both of these expressions must be equal, we have that 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 stumbling blocks can be removed with the bulldozer of foundational concepts. Study these properties and learn them. You will then be on the road to mathematical mastery.