Basic Mathematical Operations – Exponents

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Exponents comprised a juicy morsel of basic math 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 the student to be thoroughly familiar 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 is read “3 squared” or “3 raised 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, indeed, in all of mathematics, and understanding their properties and how to work with them is extremely important. Mastery of exponents requires that the student 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 pearls on a necklace. At x^3 = x*x*x, you have three x (pearls) on the string. At x^2, you have two pearls. So, in the product you have five pearls, ox^5.

law of the quotient

When you divide expressions that have the same base, you simply subtract the powers. So 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 cancel. Let’s look at a numerical example to make this completely clear. Take (5*4)/4. Since 4 appears at 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 — get 5. Now let’s multiply and divide to see if this agrees with our answer: (5*4)/4 = 20/4 = 5. Check. Therefore, this cancellation property holds. 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 and in the denominator, we can use them to cancel 3 and in the numerator to get y^2. This agrees with y^(5-3) = y^2.

Power law of a power

In an expression like (x^4)^3, we have what is known as power to power. The power of a power law states that we simplify by multiplying the powers together. So (x^4)^3 = x^(4*3) = x^12. If you think about why this is so, notice that the base of 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). We now see this as a product of the same base to 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 like (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 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.

Zero Exponent Property

Any number or variable—except 0—to the power of 0 is always 1. Thus 2^0 = 1; x^0 = 1; (x + 1)^0 = 1. To see why this is so, let’s consider the expression (x^3)/(x^3). This is clearly equal to 1, since any number (except 0) or expression over itself gives 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 obtain 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’s 3^(-2) = 1/(3^2). To see why this is so, let’s consider the expression (3^2)/(3^4). If we expand this, we get (3*3)/(3*3*3*3). Using the cancellation property, we get 1/(3*3) = 1/(3^2). Using the quotient property we have that (3^2)/(3^4) = 3^(2 – 4) = 3^(-2). Since both 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 problems in pre-algebra, algebra, and even calculus. Often times, a student’s roadblocks can be removed with the Foundational Excavator. He studies these properties and learns them. Then she will be on the path to math proficiency.

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