Review of Exponent Rules - January 28, 2014

Hello Math Students! Today's class was dedicated to solidifying our knowledge of exponent rules. We worked through various examples to ensure everyone feels comfortable applying these rules. Remember, practice is key to mastering these concepts!

Here's a summary of the key exponent rules we covered:

  • Product of Powers Rule: When multiplying powers with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$
  • Quotient of Powers Rule: When dividing powers with the same base, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$
  • Power of a Power Rule: When raising a power to another power, multiply the exponents: $(a^m)^n = a^{m \cdot n}$
  • Power of a Product Rule: When raising a product to a power, distribute the exponent to each factor: $(ab)^m = a^m b^m$
  • Power of a Quotient Rule: When raising a quotient to a power, distribute the exponent to both the numerator and the denominator: $(\frac{a}{b})^m = \frac{a^m}{b^m}$
  • Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1: $a^0 = 1$ (where $a \ne 0$)
  • Negative Exponent Rule: A negative exponent indicates a reciprocal: $a^{-n} = \frac{1}{a^n}$

Example Problems (Similar to Homework):

Let's look at a few examples similar to what you'll encounter in the homework:

  1. Simplify: $z^{-2} \cdot z^{-4} \cdot z^{6}$

    Solution: Using the product of powers rule, we add the exponents: $z^{-2 + (-4) + 6} = z^0 = 1$

  2. Simplify: $(4x^3)^{-2}$

    Solution: Using the power of a product and power of a power rules: $4^{-2} \cdot (x^3)^{-2} = \frac{1}{4^2} \cdot x^{-6} = \frac{1}{16x^6}$

  3. Simplify: $\frac{x^5 y^2}{x^{-2}}$

    Solution: Using the quotient of powers rule: $x^{5-(-2)}y^2 = x^{7}y^2$

Remember to pay close attention to the signs of the exponents and to apply the rules in the correct order. Don't hesitate to review your notes and examples from class.

Homework Assignment:

For homework, please complete the following problems from the textbook:

Pg. 326-327 #32-51 all

Good luck with your homework, and I look forward to seeing you all in the next class! Keep up the great work!