U6 Laws of Exponents Notes PDF

Unit 6 Days 3-5 Name: ____________________________ Date: __________________ Period: ______ Laws of Exponents We use powers to shorten how we represent repeated multiplication. A power has two parts, a base and an exponent. Base - the...

Unit 6 Days 3-5 Name: ____________________________ Date: __________________ Period: ______ Laws of Exponents We use powers to shorten how we represent repeated multiplication. A power has two parts, a base and an exponent. Base - the number or variable that is being multiplied repeatedly in the expanded form Exponent - specifies how many times the base will be multiplied by itself (or used as a factor) A monomial is an expression consisting of one term. A term is a single number (constant) or variable, or the product of a number (coefficient) and a variable. Previously, you learned how to add and subtract monomials by combining like terms. Ex. 3𝑥 + 2𝑥 = 5𝑥 Now you will learn how to multiply and divide! *When a term does not have a visible coefficient, its coefficient is either 1 or -1. _______________________ *When a term does not have a visible exponent, its exponent is understood to be 1. ____________________ Multiplying Powers Explore multiplying powers with the same base by completing the table below… Problem Expanded Form Simplified Expression 2 3 2 ·2 2 ℎ ·ℎ 2 3 3 3𝑎 𝑏 · 2𝑎 Product of Powers (Product Rule) To multiply powers that have the same base, add the exponents. 𝑚 𝑛 𝑥 ·𝑥 = Simplify each expression. 3 4 2 −3 6 9 3 2 6 1) 2𝑥 (𝑥 )(𝑥 ) 2) (2𝑦 )(3𝑦 ) 3) − 7𝑝 𝑞 𝑟 · 4𝑝 𝑞  Raising Power(s) to a Power Explore raising a power or product/quotient of powers to another exponent by completing the table below… Problem Expanded Form Simplified Expression 3 2 (2 ) 2 4 (𝑥 ) 2 (− 3𝑥) 2 3 (2𝑎 𝑏) 𝑥 3 (𝑦) Power of a Power (Power Rule) To find a power of a power, multiply the exponents. 𝑚 𝑛 (𝑥 ) = Power of a Product To find a power of a product, find the power of each factor and multiply. 𝑚 (𝑥 · 𝑦) = Power of a Quotient To find a power of a quotient, find the power of the numerator and the power of the denominator and divide (as long as the denominator is not 0). 𝑥 𝑚 ( 𝑦 ) = Simplify each expression. 3 6 −2 5 2 2 3 3 3 2 4) (𝑥 ) 5) 𝑥 (𝑥 ) 6) (4𝑥 𝑦) 7) 𝑎𝑏 (− 2𝑎 ) 5 4 5 4 3 4 2 3 2 3 8) 6𝑎 𝑏(2𝑎 ) 9) ( 2𝑦 ) 10) (6𝑦𝑥 ) 11) 𝑎 · (− 3𝑎 ) Dividing Powers Explore dividing powers with the same base by completing the table below… Problem Expanded Form Simplified Expression 7 𝑥 3 𝑥 7 5 𝑚𝑛 2 𝑚𝑛 8 3 81𝑎 𝑏 5 9𝑎 𝑏 Quotient of Powers (Quotient Rule) To divide powers that have the same base, subtract the exponents. 𝑚 𝑥 𝑛 = 𝑥 Simplify each expression. 5 3 7 3 𝑏 𝑥𝑦 2𝑥 𝑦 2 −3 2 12) 2 13) 2 6 14) ( ) 15) ( 𝑦 ) 𝑏 𝑥𝑦 𝑥  Zero and Negative Exponents Complete the following table… Problem Simplify using Expansion Simplify using the Quotient Rule 2 𝑤 8 𝑤 3 𝑤 5 𝑤 5 𝑥 5 𝑥 2 𝑥 2 𝑥 Negative Exponent Rule Zero Exponent Rule For every nonzero number a and integer n, For every nonzero number a, −𝑛 0 𝑎 = 𝑎 = A negative exponent is defined as the multiplicative inverse of the base, raised to the power which is opposite to the given power. In simpler terms, move the base with the negative exponent from the numerator to the denominator (or vice versa), and then make the exponent positive. Always rewrite negative exponents!! Here’s another way of deriving the Negative and Zero Exponent Rules… Decreasing the exponent by 1 is the same as dividing by 3. If you continue the pattern, you will be able to calculate both zero and negative exponents. Why can’t you use 0 as a base with zero or negative exponents? Using 0 as a base with a negative exponent would result in division by zero, which is UNDEFINED. Additionally, no matter how many times you multiply zero by itself, you get 0, which leads to contradictions… Simplify each expression. 4 −2 7 −3 3 4 0 −2 4 7 16) −𝑎 𝑏 · 4𝑎 𝑏 17) (2𝑚 ) (8𝑚 )(− 6𝑛 ) 6𝑥 𝑦 18) 6 3 2𝑥 𝑦 Mixed Practice - Simplify each expression. Follow GEMDAS when simplifying exponential expressions. Simplify inside parentheses first. Evaluate exponents (power of powers) before multiplying (product rule) or dividing (quotient rule) terms together. To be in simplest form, there should be no negative exponents in your answer. 1) 2) 3) 4) 5 5 5 4 −4 3 −7 5) 8𝑥 𝑦 · 10𝑥 𝑦 6) 10𝑥 𝑦 𝑧 7) 8) 9) 10) 12) 11) 13) 14) 15) 16) 17) 18)