Simplification Rules for Algebra using Exponents Five simplification rules for exponents will help you keep notation straight and do algebrausing exponents. Practice these five rules, and you'll be able to solve just about anyalgebra problems that contains integer exponents. We'll discuss five rules for simplifying exponential expressions. Then we'll look at a numberof examples in which we simplify exponential expressions by applying one or more ofthese rules. It is important that we use the word power in these rules in a special sense, tomean the value of an exponent. The first one is the multiplication rule. When taking the product of a number, x, and thesame factor, raised to different exponents, add the exponents. This is written as: x to the n,and then the same factor again raised to a different exponent. This equals x to the sum (n+ m). The second rule is the power to a power rule. This rule states that you have a number thatcontains an exponent, and then you raise the entire thing to a different exponent.Simplifying this expression requires taking the product of the exponents, which becomesthe power. The product to a power rule is applicable when we have two different factors and they areraised to a common exponent. We distribute the exponent over each of the factorsindividually. Let me illustrate that more clearly by giving an example. Let's say we have 2×3 to the third.Well, it should be clear that this would be the same as (2×3)(2×3)(2×3). Then we cangather up the 2s in the numerator, so we have 2×3 times 3×3. It is clear now, right? Now let's look at the fractional exponent. Let's consider the fraction to a power. In thissituation, we have one number on top and another on the bottom. We are raising each toan exponent, and then distributing the exponents. Therefore, = x^n/y^n. When you raise aratio of two integers to a power, distribute the exponent to each of the two numbers. The division and negative powers rule states that if we have x to the n / x to the m, this isthe same thing as x raised to the (n-m). This rule combines two rules we already know.When we have x to the n x to the -m, we are saying that this is equivalent to x to the (n-m). Now let's work through some examples to see if you can identify which rule to apply tosimplify and solve the equation. What is 7 to the third power (7 cubed) times 7 to the seventh power? A shared factor of 7 makes it easier to apply one of the multiplication rules. So we have 7 to the (3+7), or 7 tothe 10th. What is (4 to the 3rd) to the 5th? Here we have a power raised to another power, so we have 4 to the 3 times 5, or 4 to the 15th. What about (8*9) to the 7th? Well, we apply the product to a power rule, distributing the exponent 8 over 9 to get 8/9 and then multiplying by 9/7. And it is 8 to the 7th times 9 tothe 7th. This is an example of when scientific notation can be useful. Because this is 1.00306x10to the 13th power.
What about (2/7) to the 3rd? Here we can apply the fraction to a power rule. We have 2 to the 3rd / 7 to the 3rd, or 0.023323615. Now, 10 to the 5th power times 10 to the 3rd power equals 10 to the 5-3, which equals 10squared. This equals 100. Now let's try some slightly more challenging problems. One way to approach a problemlike this is to isolate each separate factor. The equation x to the 3rd/x to the 3rd, y to the 4th / y to the 5th, and z to the 5th/z squaredis equivalent to x to the 3-3, y to the 4-5 z to the 5-2. Negative exponents rules work here.And this equal to 1 times x to the 0, which we can simplify as 1 times y to the -1 cubed. Orif you prefer, we can write this as z cubed over y. Let us try one more example. In this example, we will first isolate each factor, and then dothe negative one at the end. So it will be the product over a power rule. To find the value of x squared multiplied by y squared, set up an equation with x squaredand y squared on top and x to the third power and y to the second-squared on the bottom.The product is x times y to the fifth power. Thus, we have the equation of x to the -5, or 1 /x to the 5th. Now try some practice problems on your own. We shall discuss one more topic: how toevaluate an exponent that is, itself, in fraction form. You can solve this problem by treating it as two separate operations. In the first, you raisethe base to a standard exponent; in the second, you take the root of the resulting number. In the example given here, we have 8 raised to the 2/3rds power. This means 8 squared,cubed root of that. Or it's equally accurate to say the third root of 8 squared. The orderdoes not matter. So let's see. The cubed route of 8 is 2 because 2 cubed is 2 times 2 times 2, or 2 to thethird power. So we would have the cubed route of 8, which is 2 to the third, squared, whichwould be equal to 4. Therefore, the cubed route of 8 is 2 squared, or 4. The square root of 64 is 8, so to find the cube root of 64, we simply cube 4, whichproduces the same number. Another example is 125 to the 4/3, which equals 625. We would take the cubed root of125, which is 5, and raise it to the 4th power. So that's 5 times 5 times 5 … which is 625. As long as you perform each step of a rational or fractional exponent separately, youshould be able to solve these problems without much difficulty. That concludes our discussion of exponent rules. What I suggest is that you simplypractice the rules a little bit, and they will become second nature to you. If you practice by
working some problems, the rules will not seem difficult.