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Basic Math Facts – Exponents

Exponents make up a juicy tidbit of basic math material. Exponents allow us to raise numbers, variables and even expressions to powers, thus achieving repeated multiplication. The exponent always present in all kinds of mathematical problems requires the student to be well acquainted with its characteristics 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 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 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 throughout 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.

The law of the product

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 beads. So in the product you have five pearls, ox^5.

Quotient law

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

The power of a power law

In an expression like (x^4)^3, we have what is known as a power to a power. The power of a power law says that we simplify by multiplying the powers together. Thus (x^4)^3 = x^(4*3) = x^12. If you are wondering why this is so, note that the base in this expression is x^4. The exponent 3 tells us to use this base 3 times. Thus we get (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 obtain x^(4 + 4+ 4) = x^12.

Distributive property

This property tells us how to simplify an expression like (x^3*y^2)^3. To simplify this, we distribute the power 3 outside the parentheses inside, multiply 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 tell us to multiply this base by itself 3 times. When you do that and then rearrange the expression using the associative and commutative properties of multiplication, you can apply the first property to get the answer.

Property of Zero Exponent

Every 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 on its own yields this result. Using our quotient property, we see that this is equal to x^(3 – 3) = x^0. Since both expressions must give the same result, we have that x^0 = 1.

Property of negative exponent

When we raise a number or variable to a negative integer, it ends with u reciprocal. It is 3^(-2) = 1/(3^2). To see 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, you 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 removed with the bulldozer of foundational concepts. Study these properties and learn. Then you will be on the road to mathematical mastery.

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