Simplifying Exponents Lecture

Basic Properties of Exponents

Multiplication of Exponents

• Rule: When multiplying exponents with a common base, add the exponents.
  • Example: ( x^4 \times x^5 = x^{4+5} = x^9 )
  • Explanation: You are multiplying four ( x ) variables with five ( x ) variables, totaling nine ( x ) variables.

Division of Exponents

• Rule: When dividing exponents with a common base, subtract the exponents.
  • Example: ( \frac{x^7}{x^3} = x^{7-3} = x^4 )
  • Explanation: Cancel three ( x ) variables from both numerator and denominator, leaving four.

Raising an Exponent to Another Exponent

• Rule: Multiply the exponents.
  • Example: ( (x^3)^4 = x^{3 \times 4} = x^{12} )
  • Explanation: Each ( x^3 ) is three ( x ) variables, so ( x^{12} ) results from multiplying four sets of three.

Special Exponent Values

Zero Exponent Rule

• Rule: Any non-zero number raised to the 0 power is 1.
  • Example: ( x^0 = 1 ), ( 4^0 = 1 )

Negative Exponent Rule

• Rule: A negative exponent indicates a reciprocal.
  • Example: ( x^{-3} = \frac{1}{x^3} )
  • Explanation: Move the variable to the denominator and make the exponent positive.

Simplifying Expressions

Example Problems

1. Simplifying Multiplication with Common Bases:
   • ( x^5 = 1/x^5 ) if negative.
   • ( (x^3)^5 = x^{15} )
2. Division Example:
   • ( \frac{x^7}{x^{12}} = \frac{1}{x^5} )

Larger Expression Simplification

• Example: Simplifying ( (3x^2)^3 ) results in:

  • ( 3^3 \times x^6 = 27x^6 )

• Complex Example:

  • ( -2x^3y^4 ) raised to the power of 2 results in:
  • ( (-2)^2 \times x^6 \times y^8 = 4x^6y^8 )

Multiplication and Division of Complex Expressions

Multiplying and Simplifying

• Example: ( 5x^3 \times 4x^7 = 20x^{10} )
• Example: ( 7x^6 \times 5x^4 = 35x^{10} )

Complex Expressions with Division

• Example:
  • ( \frac{24x^7y^3}{8x^4y^{-2}} = 3x^3y^{15} )

Final Complex Example

• Example:
  • Simplifying ( \frac{35x^3y^5}{63x^4y^7} ) raised to 2 results in:
  • Final result: ( \frac{25x^{14}}{81y^4} )

Key Pointers

• Always apply zero-exponent and negative-exponent rules correctly.
• Distribute powers over multiplication and division.
• Simplify within the parentheses before applying powers.

These notes summarize the key points discussed in the lecture on simplifying exponents, covering multiplication, division, and raising powers, as well as handling special cases like zero and negative exponents.