Simplification Rules for Algebra using Exponents
Five simplification rules for exponents will help you keep notation straight and do algebra using exponents. Practice these five rules, and you’ll be able to solve just about any algebra problems that contains integer exponents. We’ll discuss five rules for simplifying exponential expressions. Then we’ll look at a number of examples in which we simplify exponential expressions by applying one or more of these rules. It is important that we use the word power in these rules in a special sense, to mean the value of an exponent. The first one is the multiplication rule. When taking the product of a number, x, and the same 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 that contains an exponent, and then you raise the entire thing to a different exponent. Simplifying this expression requires taking the product of the exponents, which becomes the power. The product to a power rule is applicable when we have two different factors and they are raised to a common exponent. We distribute the exponent over each of the factors individually. The product to a power rule is applicable when we have two different factors and they are raised to a common exponent. We distribute the exponent over each of the factors individually. 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 can gather 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 this situation, we have one number on top and another on the bottom. We are raising each to an exponent, and then distributing the exponents. Therefore, = x^n/ y”n. When you raise a ratio 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 is the same thing as × 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 to simplify 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 to the 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? exponent 8 over 9 to get Well, we apply the product to a power rule, distributing the 89 and then multiplying by 9/7. And it is it 8 to the 7th times 9 to the 7th. This is an example of when scientific notation can be useful. Because this is 1.00306×10 to 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 10 squared. This equals 100. Now let’s try some slightly more challenging problems. One way to approach a problem like 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 squared is 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. Or if you prefer, we can write this as cubed over y. Let us try one more example. In this example, we will first isolate each factor, and then do the 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 squared and 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 to evaluate an exponent that is, itself, in fraction form. You can solve this problem by treating it as two separate operations. In the first, you raise the 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 order does 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 the third power. So we would have the cubed route of 8, which is 2 to the third, squared, which would 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, which produces the same number. Another example is 125 to the 4/3, which equals 625. We would take the cubed root of 125, 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, you should be able to solve these problems without much difficulty. That concludes our discussion of exponent rules. What I suggest is that you simply practice the rules a little bit, and they will become second nature to you. If you practice by Exponent Simplification Rules 1.Multiplication Rule x^n x^m=x^(n+m) 2.Power to a Power (x^n)^m=x^nm 3.Product to a Power (2*3)^3(2*3)(2*3)(2*3)2^3 3^3 (xy)^n=x^n y^n 4.Fraction to a Power (x/y)^n=(x^n)/(y^n) 5.Division and Negative Powers x^n/x^m=x^(n-m) x^n x^-m=x^(n-m) Examples (7^3)(7^7)=7^(3+7)=7^10 (x^3y^4z^5)/(x^3y^4z^2)=(x^3)/(x^3)(y^4)/(y^4)(z^5)/(z^2) working some problems, the rules will not seem difficult.
