1-3 2 Laws of Exponents
The 1-3 2 Laws of Exponents are fundamental rules that help simplify expressions involving exponents. These laws are essential for solving various mathematical problems, especially in algebra.
1. Product of Powers Rule
The Product of Powers Rule states that when multiplying two expressions with the same base, you add the exponents. Mathematically, this is represented as:
\[ a^m \cdot a^n = a^{m+n} \]
Example: Simplify \( 2^3 \cdot 2^4 \).
Solution: Using the Product of Powers Rule, \( 2^3 \cdot 2^4 = 2^{3+4} = 2^7 \).
2. Power of a Power Rule
The Power of a Power Rule states that when raising a power to another power, you multiply the exponents. Mathematically, this is represented as:
\[ (a^m)^n = a^{m \cdot n} \]
Example: Simplify \( (3^2)^3 \).
Solution: Using the Power of a Power Rule, \( (3^2)^3 = 3^{2 \cdot 3} = 3^6 \).
3. Power of a Product Rule
The Power of a Product Rule states that when raising a product to a power, you raise each factor to that power. Mathematically, this is represented as:
\[ (a \cdot b)^n = a^n \cdot b^n \]
Example: Simplify \( (4 \cdot 5)^2 \).
Solution: Using the Power of a Product Rule, \( (4 \cdot 5)^2 = 4^2 \cdot 5^2 = 16 \cdot 25 = 400 \).
Examples and Analogies
Understanding these laws can be made clearer through analogies:
Analogy: Think of exponents as layers of a cake. When you add layers (exponents) to the same cake (base), you are increasing the height of the cake. When you multiply layers (exponents) of different cakes (bases), you are making each cake taller individually.
By mastering the 1-3 2 Laws of Exponents, you can simplify complex expressions and solve problems more efficiently.