How do you multiply exponents with variables?

To multiply exponents with variables, you add the exponents of like bases.

Here's a breakdown of the process, along with examples and insights from the provided reference:

Understanding the Basics

When you have variables with exponents, it means that the variable is multiplied by itself a certain number of times. For instance, x³ means x x x. When multiplying these terms together, you're effectively combining these multiplications, which means we add the exponents of the same base.

The Rule:

The rule for multiplying exponents with variables is:

*x^m xⁿ = x^m+n**

Where:
- x is the base (the variable).
- m and n are the exponents.

Step-by-Step Guide with Examples

Here's how to apply the rule, using the information from the reference provided:

Identify like bases: Make sure the variables (bases) being multiplied are the same.
Add the exponents: Add the exponents of the like bases.
Keep the base the same: The base remains unchanged in the final result.

Examples:

Example 1:
- x² * x³ = x²⁺³ = x⁵
- Here, we have two x's multiplied by three x's, resulting in five x's multiplied together (x x x x x).
Example 2 (as highlighted in the reference):
- x x² x⁵ = x¹ x² x⁵ = x¹⁺²⁺⁵ = x⁸
- As the video mentioned, "all these x's are multiplied together." Thus, 8 x's are present when multiplied together.

Practical Insights:

Implicit Exponents: Remember that if a variable does not have a visible exponent, it has an implied exponent of 1 (e.g., x is x¹).
Coefficients: If you have coefficients, multiply them as usual, and then apply the exponent rule for the variables (e.g. 2x² * 3x³ = 6x⁵).

Table: Multiplication of Exponents with Variables

Expression	Result	Explanation
x⁴ * x²	x⁶	Add the exponents 4 + 2 = 6
a * a³	a⁴	a¹ * a³ = a¹⁺³ = a⁴
y⁵ * y	y⁶	y⁵ * y¹ = y⁵⁺¹ = y⁶
2b² * 3b⁴	6b⁶	2 * 3 = 6, then add the exponents for b: 2 + 4 = 6

Conclusion

Multiplying exponents with variables involves adding the exponents of like bases while keeping the base unchanged. Understanding this rule helps simplify and solve algebraic expressions efficiently, as demonstrated by the examples and the video reference.

askvity