(x%5Eb).jpeg "(x^a)(x^b)")
Understanding the Multiplication of Exponents: (x^a)(x^b)
When multiplying exponents with the same base, we can simplify the expression using a simple rule. This rule states that we add the exponents while keeping the base the same.
The Rule
** (x^a)(x^b) = x^(a+b)**
This rule is based on the fundamental understanding of exponents. For example:
- x^a means multiplying x by itself 'a' times.
- x^b means multiplying x by itself 'b' times.
Therefore, (x^a)(x^b) is simply multiplying x by itself (a+b) times.
Example:
Let's take a simple example:
(x^3)(x^2) = x^(3+2) = x^5
- x^3 is equal to x * x * x
- x^2 is equal to x * x
- Therefore, (x^3)(x^2) is equal to (x * x * x) * (x * x) = x^5
Applying the Rule:
This rule can be applied to any exponents with the same base, whether they are integers, fractions, or even variables. For example:
- (y^2)(y^5) = y^(2+5) = y^7
- (a^(1/2))(a^(3/2)) = a^(1/2 + 3/2) = a^2
- (m^n)(m^p) = m^(n+p)
Conclusion:
The rule for multiplying exponents with the same base is a fundamental concept in algebra. Understanding this rule allows us to simplify complex expressions and work with exponents more efficiently.