Lecture on Simplifying Exponents

Basic Properties of Exponents

Addition of Exponents:
- When multiplying with the same base, add the exponents.
- Example: ( x^4 \times x^5 = x^{4+5} = x^9 ).
Subtraction of Exponents:
- When dividing with the same base, subtract the exponents.
- Example: ( \frac{x^7}{x^3} = x^{7-3} = x^4 ).
Power of a Power:
- When raising an exponent to another power, multiply the exponents.
- Example: ( (x^3)^4 = x^{3 \times 4} = x^{12} ).
Zero Exponent Rule:
- Any number raised to the power of zero is 1.
- Example: ( x^0 = 1 ).
Negative Exponents:
- A negative exponent indicates reciprocal.
- Example: ( x^{-3} = \frac{1}{x^3} ).

Example 1:
- ( -3^2 ) vs ( (-3)^2 )
  - ( -3^2 = -9 ) because the negative is not squared.
  - ( (-3)^2 = 9 ) because both the negative and the base are squared.
Example 2:
- Simplify ( (3x^2)^3 )
  - Distribute the power: ( 3^3 \times x^{2 \times 3} = 27x^6 ).
Example 3:
- Simplify ( (-2x^3y^4)^2 )
  - Distribute the power: ( (-2)^2 \times x^{3 \times 2} \times y^{4 \times 2} = 4x^6y^8 ).

Problem 1: Multiply ( 5x^3 \times 4x^7 )
- Solution: ( 5 \times 4 \times x^{3+7} = 20x^{10} ).
Problem 2: Simplify ( \frac{24x^7y^3}{8x^4y^{-2}} )
- Solution:
  - Divide coefficients: ( 24 / 8 = 3 ).
  - Subtract exponents: ( x^{7-4} = x^3 ), ( y^{3-(-2)} = y^5 ).
  - Final answer: ( 3x^3y^5 ).
Problem 3: Simplify ( \frac{35x^3y^5}{63x^4y^7} ) squared
- Simplification:
  - Pre-simplify: ( \frac{35}{63} = \frac{5}{9} ).
  - Exponents: ( x^{3-4} = x^{-1} ) and ( y^{5-7} = y^{-2} ).
  - Move negative exponents: ( \frac{5}{9x^1y^2} ).
  - Square the expression: ( \left(\frac{5}{9x^1y^2}\right)^2 = \frac{25}{81x^2y^4} ).