Unit 1 - Lesson 1: Power of a Power and Exponent Laws
Master advanced algebraic exponent rules: power of a power, distributing powers across multiplication and division, and simplifying complex algebraic expressions.
1. Fundamental Laws of Exponents in Real Numbers
Product & Quotient of Same Bases
Multiplying like bases: Add the exponents → a^m × a^n = a^(m+n)
Dividing like bases: Subtract the exponents → a^m ÷ a^n = a^(m-n)
Zero & Negative Exponents
Zero exponent: Any non-zero real number to power 0 equals 1 → a^0 = 1
Negative exponent: Represents the reciprocal → a^(-n) = 1 / a^n
Core Rule: Power of a Power Law:
For any non-zero real number a and any integers m, k:
When raising an exponential expression to another power, multiply the exponents directly.
2. Power of a Product & Power of a Quotient
Power of a Product
Example: (3^4 × 5^2)^3 = 3^12 × 5^6
Power of a Quotient
Example: (x^2 / y^3)^4 = x^8 / y^12
3. Interactive Worked Exercises
Solution Steps:
1) Inside parentheses: (3 + 3 + 3) = 3 × 3 = 9 = 3^2
2) Raise to power 4: (3^2)^4 = 3^(2 × 4) = 3^8
Final Answer = 3^8
Solution Steps:
1) Simplify inside parentheses by subtracting exponents:
→ x^(2 - 4) = x^(-2) = 1 / x^2
→ y^(5 - 2) = y^3
→ Simplified base = y^3 / x^2
2) Apply power of quotient law: (y^3 / x^2)^3 = y^(3 × 3) / x^(2 × 3)
Final Answer = y^9 / x^6
4. Interactive Assessment Quiz
Choose the correct answer for each question, then click "Submit Answers":
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