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Monomials, and Negative Exponents Multiplying and Dividing Monomials Monomials A monomial is an expression in algebra that contains one term, like 3xy. Monomials include: numbers, whole numbers and variables that are multiplied together, and variables that are multiplied together. Identifying a Monomial Any number, all by itself, is a monomial, like 5 or 2700. A monomial can also be a variable, like “m” or “b”. It can also be a combination of these, like 98b or 78xyz. It cannot have a fractional or negative exponent. These are not monomials: 45x+3, 10 - 2a, or 67a - 19b + c. Two rules about monomials are: •A monomial multiplied by a monomial is also a monomial. •A monomial multiplied by a constant is also a monomial. Multiplying Monomials Recall that exponents are used to show repeated multiplication. The power of 3 23 = 2 ∙ 2 ∙ 2 Three factors Use the definition of an exponent to find the rule for multiplying powers with the same base. 3 factors 4 factors 23 ∙ 24 = (2 ∙ 2 ∙ 2) ∙ (2 ∙ 2 ∙ 2 ∙ 2) = 27 7 factors Product of Powers To multiply powers with the same base, keep the base and add their exponents am ∙ an = am+n 32 ∙ 34 = 32+4 = 36 Products of Powers Examples 53 ∙ 54 = 57 5 3+4 = 7 (4n3)(2n6) = (4 ∙ 2)(n3 ∙ n6) Keep the base Add the exponents separate the non-exponent factors from the exponent factor – use the Commutative and Associative Properties. = (8)(n3 ∙ n6) The common base is n = (8n9) add the exponents Dividing Monomials Recall dividing it the opposite of multiplying You can also write a rule for finding quotients of powers 26 2∙2∙2∙2∙2∙2 = 22 2∙2 1 1 2∙2∙2∙2∙2∙2 = 2∙2 = 24 1 1 6 factors 2 factors Divide out the common factors Quotient of Powers To divide powers with the same base, keep the base and subtract their exponents am m - n a cannot = 0 = a an 45 5 - 2 = 43 = 4 42 Products of Powers Examples 57 = 54 3 5 57- 3=4 4n9 = (4 ÷ 2)(n9 ÷ n6) 6 2n Keep the base Subtract the exponents separate the non-exponent factors from the exponent factor n9 = (2)( 6) n The common base is n = (2n3) subtract the exponents Do some examples Do some examples Negative Exponents What Is A Negative Exponent? 8-2 That exponent is negative ... what does it mean? A Positive exponent meant to multiply. So Negative? Must be the opposite of multiplying. Dividing! Dividing is the inverse (opposite) of Multiplying. HOW? A negative exponent means how many times to divide by the number. Example: 8-1 = 1 ÷ 8 = 1/8 = 0.125 Or Example: 2-3= 1 ÷ 2 ÷ 2 ÷ 2 = 0.125 Notice that a negative exponents means you divide by 1 and then divide the base the amount of times as the exponents states. There a easier way to look at this! 2-3= 1 ÷ 2 ÷ 2 ÷ 2 = 1 ∕8 = 0.125 What did I do? 1st Calculate the positive exponent 2-3 → 2 ∙ 2 ∙ 2 = 8 Then Then take the Reciprocal (i.e. 1/an) = 2-3 1 1 = 3 = = 0.125 2 8 Negative Exponents Any nonzero number to the negative n power is the multiplicative inverse of its nth power. 1 = 𝑛 a cannot = 0 𝑎 1 1 -4 5 = 4= = 0.0016 625 5 a-n Understanding Write each expression using a positive exponent. Write each expression using a negative exponent. 1 = 2 6 1 -5 x = 5 𝑥 1 1 = 2 = 3-2 9 3 6-2 5-6 = p-4 = 1 25 = 1 𝑑5 = d-5 1 16 1 52 = = 5-2