Exponents comprise a juicy tidbit of basic-math-facts material. Exponents allow us to raise numbers, variables, and already expressions to powers, consequently achieving repeated multiplication. The ever present exponent in all kinds of mathematical problems requires that the student be thoroughly conversant with its features and similarities. 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 strength,” 3 is the base and 2 is the strength 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 average x~x~x. In (x + 1)^2, this method (x + 1)*(x + 1). Exponents are omnipresent in algebra and indeed all of mathematics, and understanding their similarities and how to work with them is extremely important. Mastering exponents requires that the student be familiar with some basic laws and similarities.
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. consequently in the product you have five pearls, or x^5.
When dividing expressions involving the same base, you simply subtract the powers. consequently in (x^4)/(x^2) = x^(4-2) = x^2. Why this is so depends on the cancellation character of the real numbers. This character 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 average—to get 5. Now let’s multiply and divide to see if this agrees with our answer: (5~4)/4 = 20/4 = 5. Check. consequently this cancellation character 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.
strength of a strength Law
In an expression such as (x^4)^3, we have what is known as a strength to a strength. The strength of a strength law states that we simplify by multiplying the powers together. consequently (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. consequently we would acquire (x^4)*(x^4)*(x^4). Now we see this as a product of the same base to the same strength and can consequently use our first character to get x^(4 + 4+ 4) = x^12.
This character tells us how to simplify an expression such as (x^3~y^2)^3. To simplify this, we spread the strength 3 outside parentheses inside, multiplying each strength 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 similarities of multiplication, you can then apply the first character to get the answer.
Zero Exponent character
Any number or variable—except 0—to the 0 strength is always 1. consequently 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 consequence. Using our quotient character, we see this is equal to x^(3 – 3) = x^0. Since both expressions must provide the same consequence, we get that x^0 = 1.
Negative Exponent character
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 acquire (3~3)/(3~3~3~3). Using the cancellation character, we end up with 1/(3~3) = 1/(3^2). Using the quotient character 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 similarities of exponents will give students the substantial foundation they need to tackle all kinds of pre-algebra, algebra, and already calculus problems. Often times, a student’s stumbling blocks can be removed with the bulldozer of foundational concepts. Study these similarities and learn them. You will then be on the road to mathematical expert.