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Basic Math Facts – Exponents
Exhibitors include a juicy treat of basic math material. Exponents allow us to raise numbers, variables, and even expressions to powers, thus performing repeated multiplication. The ever-present exponent in all kinds of mathematical problems requires the student to be fully aware of its characteristics and properties. Here we consider the laws whose knowledge will allow any student to master this subject.
In the expression 3^2, which reads “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 goes for variables and variable expressions. In x^3, it means x*x*x. In (x + 1)^2, it means (x + 1)*(x + 1). Exponents are ubiquitous in algebra and indeed all mathematics, and it is extremely important to understand their properties and know how to use them. Mastering exponents requires the student to 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 beads on a string. In x^3 = x*x*x, you have three x’s (beads) on the string. In x^2, you have two pearls. So in the product you have five pearls, or x^5.
law of the quotient
When you divide expressions involving the same base, you are simply subtracting the powers. So in (x^4)/(x^2) = x^(4-2) = x^2. Why this is so depends on the undo property real numbers. This property indicates that when the same number or the same variable appears in both the numerator and the denominator of a fraction, then this term can be canceled. Let’s take a numerical example to clarify all this. Take (5*4)/4. Since 4 appears both at 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 it matches our answer: (5*4)/4 = 20/4 = 5. Check. So this cancellation property holds. In an expression like (y^5)/(y^3), it’s (y*y*y*y*y)/(y*y*y), if we expand. Since we have 3 y’s in the denominator, we can use these 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 called a power to a power. The power of a power law says we simplify by multiplying the powers together. So (x^4)^3 = x^(4*3) = x^12. If you wonder why this is so, note that the base of this expression is x^4. Exponent 3 tells us to use this base 3 times. We would thus obtain (x^4)*(x^4)*(x^4). Now we see this as a product of the same base to the same power and so can 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 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, note 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 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 power of 0 is always 1. So 2^0 = 1; x^0 = 1; (x + 1)^0 = 1. To understand why this is so, consider the expression (x^3)/(x^3). This is clearly equal to 1, since any number (except 0) or expression on itself gives this result. Using our quotient property, we see that this equals x^(3 – 3) = x^0. Since both expressions must give 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 understand why this is so, consider 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 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 give students the solid foundation they need to tackle all kinds of pre-algebra, algebra, and even calculus problems. Often a student’s stumbling blocks can be eliminated with the bulldozer of fundamental concepts. Study these properties and learn them. You will then be on the way to mathematical mastery.
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