UNIT 5
Polynomials and Properties of Exponents
LESSON 5
Exponents and Division
One property of exponents that is important to understand when dividing with exponents is the quotient rule. This rule states that when dividing two terms with the same base, you can subtract the exponents.
For example, if you are dividing x^3 by x^2, you can use the quotient rule to simplify as follows:
x^3 / x^2 = x^(3-2) = x^1 = x
Another important concept to understand is how negative exponents work. Remember that a negative exponent means you are taking the reciprocal of the base raised to the positive exponent.
For example, if you have x^-2, this equals 1 over x^2. And if you have (1/x)^-3, this equals x^3.
When dividing with negative exponents, it can be helpful to first rewrite the expression so that all exponents are positive. Then, you can apply the quotient rule.
For example, if you have x^-2 / y^-3, you can rewrite this as (y^3 / x^2) and then use the quotient rule as follows:
x^-2 / y^-3 = (y^3 / x^2) = y^(3-(-2)) / x^(2-(-3)) = y^5 / x^5
Overall, when dividing with exponents, it's important to keep track of the base and the exponent of each term and use the quotient rule and/or understanding of negative exponents as necessary.
x^9/x^2
Using the quotient rule, we can simplify x^9/x^2 as follows:
x^9 / x^2 = x^(9-2) = x^7
Therefore, x^9/x^2 simplifies to x^7.
In this lesson, we will be exploring how exponents and division are related. We will learn about the properties of exponents that help us simplify expressions and solve problems involving division.
To understand exponents and division, let's review the basics:
1. Exponents: An exponent tells us how many times a base number is multiplied by itself. For example, in the expression 2^3, the base is 2, and the exponent is 3. This means that we multiply 2 by itself 3 times: 2 * 2 * 2 = 8.
2. Division: Division is the process of splitting a number into equal parts or groups. It is denoted by the division symbol (/). For example, in the expression 8 / 2, we divide 8 into 2 equal parts, which gives us 4.
Now, let's see how exponents and division are connected:
Property 1: When dividing two numbers with the same base, subtract the exponents. For example, if we want to simplify the expression (3^4) / (3^2), we subtract the exponents: 4 - 2 = 2. So, the simplified expression is 3^2 = 9.
Property 2: When dividing numbers with different bases but the same exponent, divide the bases. For example, if we want to simplify the expression (2^3) / (4^3), we divide the bases: 2 / 4 = 1/2. So, the simplified expression is (1/2)^3.
Property 3: When dividing numbers with different bases and exponents, first simplify each base and exponent separately, and then divide them. For example, if we want to simplify the expression (2^3) / (3^2), we simplify each part individually: 2^3 = 8 and 3^2 = 9. Then, we divide them: 8 / 9.
These properties help us work with exponents and division in a systematic way, allowing us to simplify expressions and solve problems more easily. It is important to understand these properties and practice using them to become comfortable with exponents and division.