Lecture on Negative Exponents

Key Concepts

• Negative Exponent Rule:

  • ( x^{-n} = \frac{1}{x^n} )
  • Useful for transforming expressions with negative exponents into those with positive exponents.

• Simplifying Expressions:

  • Example 1: ( x^{-2} ) becomes ( \frac{1}{x^2} ).
  • Example 2: ( \frac{1}{y^{-3}} ):
    • Simplify to ( y^3 ) by moving ( y^{-3} ) to the numerator and changing the exponent to positive.

Manipulating Fractional Powers

• Reciprocal Understanding:

  • Dividing by a fraction is equivalent to multiplying by its reciprocal.

• Negative Exponent in Fractions:

  • Move negative exponent factor to the opposite side of the fraction bar to make the exponent positive.

Examples

• Example with Coefficients and Variables:
  • Simplifying: ( \frac{2}{8}x^{-3}z^{-2}y^5 )
    • Simplify numbers: ( \frac{1}{4} )
    • Move ( x^{-3} ) to the denominator: ( x^3 )
    • Move ( z^{-2} ) to the numerator: ( z^2 )
    • Final simplification: ( \frac{z^2 y^5}{4x^3} )

Power of a Power Rule

• Example: ( (2x^{-2}y^3)^{-3} )
  • Distribute (-3) across terms:
    • ( 2^{1*(-3)} ), ( x^{2*(-3)} ), ( y^{3*(-3)} )
  • Result: ( \frac{x^6}{8y^9} )
  • Move terms with negative exponents.*

Quotient Rule for Exponents

• Simplification Method:
  • Reduce coefficients: ( \frac{50}{12} = \frac{25}{6} )
  • Subtract exponents when dividing:
    • For ( x ): ( -7 - 4 = x^{-11} )
    • For ( y ): ( -2 - (-3) = y^1 )
  • Final Answer: ( \frac{25y}{6x^{11}} )

Conclusion

• Focus on ensuring positive exponents in the final answer.
• Techniques to simplify expressions efficiently using the rules of exponents.

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