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Laws of Exponents Tammy Wallace What is a monomial? Mono means one. A monomial is an expression that is a number, a variable, or the product of a number and one or more variables. Examples: 12 𝑦 number variable 𝑐 −5𝑥 𝑦 3 Product of a number and variables 2 What about more than one? Bi means two Tri means three Poly means more than one. A polynomial is a monomial or the sum or difference of two or more monomials. REMEMBER: An expression is NOT a polynomial if there is a variable in the denominator. Determine whether each expression is a polynomial. If it is, state what type of polynomial it is. 7𝑦 − 3𝑥 + 4 A trinomial 10𝑥 3 𝑦𝑧 2 A monomial 5 + 7𝑦 2 2𝑦 NOT a polynomial because of the variable in the denominator Some monomials have exponents. For example, 𝑥 2 is read as x to the second power. X would be the base of the term and 2 would be the exponent. State the base and exponent for each monomial. 34 3 The base is _____ 4 The exponent is _____ 𝑦3 y The base is _____ 3 The exponent is _____ What are exponents? Exponents means to multiply a number by itself by on the number of the exponent. For example, 4³ means to multiply four by itself 4 times. 4 ∙ ______ 4 ∙ ______ 4 = ______. 64 So, 4³ = ______ Note: This is not the same as 4 ∙ 3 which equals 12. Expand the following monomials and simplify to a number if possible. 34 𝑦3 𝟑 ∙ 𝟑 ∙ 𝟑 ∙ 𝟑 = 𝟖𝟏 𝑦∙𝑦∙𝑦 Multiplying Monomials There are two types of rules to remember when multiplying monomials. 1. Multiply Rule 2. Power to a Power Rule Multiplying Rule: When multiplying monomials, multiply the coefficients and add the exponents. Keep in mind, a monomial can be a number, a variable, or the product of numbers and variables. Multiplying Rule: When multiplying monomials, multiply the coefficients and add the exponents • When multiplying monomials, always analyze what is given and apply the rule based that. NUMBERS VARIABLES (𝑥 2 )( 𝑥 3 ) What part of the rule needs 𝟖 2 (4) = _______ to be applied? Adding the exponents. PRODUCT OF NUMBERS AND VARIABLES (2𝑎2 )(3𝑎5 ) What part of the rule should be applied? Multiplying coefficients AND adding exponents. (𝑥 2 )( 𝑥 3 ) 𝟔 2 ∙ 3 = ____ 𝟐 𝟑 = (𝑥 _______+_______ ) 𝟔 = (______𝑎2 )( 𝑎5 ) = 𝟓 𝒙 𝟐 𝟓 𝟔 _________+_________ = (_____𝑎 ) = 𝟔𝒂𝟕 Multiplying Rule: When multiplying monomials, multiply the coefficients and add the exponents NUMBERS VARIABLES (𝑚𝑛)( 𝑚3 𝑛)(𝑚4 𝑛5 ) 𝟕𝟓 𝟐𝟓 ∙ 𝟑= ___ Apply the same rule as above but only to add exponents to like variables. (𝑚𝑛)( 𝑚3 𝑛)(𝑚4 𝑛5 ) 𝟑 𝟏 𝟒 _______+______+________ 𝟏 𝟏 𝟓 ______+_______+_______ 𝑚 𝑛 𝟖 𝟕 =𝒎 𝒏 PRODUCT OF NUMBERS AND VARIABLES (𝑥 3 )(4𝑥𝑦 5 ) What part of the rule should be applied? Multiplying coefficients AND adding exponents. 𝟒 1 ∙ 4 = ____ 𝟑 𝟏 𝟓 𝟒 _______+_______ _______ =(_____𝑥 𝑦 ) = 𝟒𝒙𝟒 𝒚𝟓 Multiplying Monomials When multiplying monomials, the exponents are added together. However, if any exponent(s) are located outside the parenthesis, everything inside the parenthesis has to be raised to that outside power. NUMBERS PRODUCT OF NUMBERS AND VARIABLES VARIABLES 𝟐 3(5)² = 3(__)(___) 5 5 = 75 𝒙 𝟒 𝒚𝟑 What exponent is outside the 𝟐 parenthesis? __ inside the Expand the monomial ______ parenthesis by the exponent outside directly _________the parenthesis. 𝟒 𝟒 𝟑 𝟒 𝟒 𝟑 = (𝑥 _____ )(𝑥 _____ )𝑦 ______ = 𝑥 _____+_______ 𝑦 _____ 𝟖 𝟑 =𝒙 𝒚 𝟐 𝟑𝒙𝟒 𝟐 Remember to ONLY expand inside the the monomial ___________ parenthesis by the exponent directly outside ______the parenthesis. 4) 𝟒)(_____𝑥 𝟑 ____ 𝟑 ____ = 2(_____𝑥 𝟒 𝟒 ) 𝟑 ∙ ___𝑥 𝟑 ____+_____ = 2(____ =2∙ 𝟗 𝟖 _____𝑥 ____ 𝟖 𝟏𝟖𝒙 = _________ 2 4 2 Multiply −7𝑥 𝑦 3𝑥𝑦 = −7𝑥 2 𝑦 4 (−7𝑥 2 𝑦 4 ) 3𝑥𝑦 2 = −7 ∙ −7𝑥 2 𝑦 4 (𝑥 2 𝑦 4 ) 3𝑥𝑦 2 = 49𝑥 2 𝑦 4 (𝑥 2 𝑦 4 ) 3𝑥𝑦 2 = 49 · 3𝑥 2+2+1 𝑦 4+4+2 = 147𝑥 𝑦 5 10 2