UNIT 5
Polynomials and Properties of Exponents
LESSON 3
Exponents and Multiplication
In this lesson, we will learn about exponents and how to multiply numbers with exponents. Here are step-by-step explanations of the main concepts covered in Lesson 3:
1. Exponents: An exponent represents the number of times a base number is multiplied by itself. It is written as a superscript after the base number. For example, in the expression 2^3, the base is 2 and the exponent is 3. It means 2 is multiplied by itself three times, which equals 8.
2. Multiplying numbers with exponents: When multiplying numbers with exponents, you can either multiply the base numbers and add the exponents, or keep the base numbers separate and add the exponents. Here are the two methods:
Method 1: Multiply the base numbers and add the exponents.
Example: 2^3 * 2^4 = (2 * 2)^(3 + 4) = 2^7 = 128
Method 2: Keep the base numbers separate and add the exponents.
Example: 2^3 * 2^4 = 2^(3 + 4) = 2^7 = 128
3. Power of a product rule: When a product is raised to an exponent, you can distribute the exponent to each factor inside the parentheses. Here's the rule:
(a * b)^n = a^n * b^n
Example: (2 * 3)^4 = 2^4 * 3^4 = 16 * 81 = 1296
4. Simplifying expressions with exponents: Sometimes you may have to simplify expressions with exponents using the rules mentioned above. Just remember to follow the given order of operations, which is parentheses, exponents, multiplication and division (from left to right), and finally addition and subtraction (from left to right).
Example: Simplify the expression 3^2 * 3^3 / 3^4.
Solution: Using the rules, we can rewrite the expression as (3 * 3)^(2 + 3) / 3^4 = 3^5 / 3^4 = 3^(5 - 4) = 3^1 = 3.
Remember to practice these concepts by solving more examples and exercises.
In this lesson, we will be focusing on exponents and how they relate to multiplication. Specifically, we will explore the following topics:
1. Product rule of exponents: This rule states that when multiplying two terms with the same base, you can add their exponents. For example, x^2 * x^3 = x^(2+3) = x^5. We can use this rule to simplify expressions with exponents.
2. Power of a product rule: This rule states that when raising a product to an exponent, each term within the parentheses should be raised to that exponent. For example, (ab)^2 = a^2 * b^2. This rule is helpful when dealing with expressions involving multiple variables.
3. Simplifying expressions with exponents: We will learn how to simplify expressions by combining like terms and applying the product rule and power of a product rule.
To understand and apply these concepts, it's important to have a solid foundation in multiplication and basic exponent rules. If you are not familiar with these concepts, it would be helpful to review them before continuing with this lesson.
In this lesson, we will provide examples and explanations of each rule and guide you through practice problems to reinforce your understanding of exponents and multiplication.
In Lesson 3, we will explore the properties of exponents when it comes to multiplication.
Recall that an exponent is a shorthand way of writing the repeated multiplication of a base number. For example, 2³ means 2 multiplied by itself three times: 2 x 2 x 2 = 8.
When we multiply two numbers with exponents, we can use the following rules:
Product Rule: When multiplying two numbers with the same base, we can add their exponents.
For example:
2² x 2³ = 2^(2+3) = 2^5 = 32
This means that 2² x 2³ is the same as 2 to the power of 5.
Quotient Rule: When dividing two numbers with the same base, we can subtract their exponents.
For example:
4⁵ / 4³ = 4^(5-3) = 4² = 16
This means that 4⁵ divided by 4³ is the same as 4 to the power of 2.
Power Rule: When raising a number with an exponent to another exponent, we can multiply the exponents.
For example:
(2⁴)³ = 2^(4x3) = 2¹² = 4096
This means that (2⁴)³ is the same as 2 to the power of 12.
These rules can also be applied to variables with exponents. For instance, when multiplying two variables with the same base, we can add their exponents.
If we have x² times x³, we can add the exponents to get x^(2+3) = x^5.
Similarly, when dividing two variables with the same base, we can subtract their exponents.
If we have y^7 divided by y^4, we can subtract the exponents to get y^(7-4) = y^3.
When raising a variable with an exponent to another exponent, we can multiply the exponents.
If we have (z^2)^3, we can multiply the exponents to get z^(2x3) = z^6.
By understanding these rules, we can simplify expressions with exponents and make calculations easier and quicker.