# 0028 Exponents & Polynomials Notes

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Page 1 of 10 10.1 -Exponents Exponent Review Exponents mean repeated multiplication 3 5 = 3 3 3 3 3 Any base with an exponent of 0 equals 1 𝑚𝑚 0 = 1 7 0 = 1 Exponents & negative bases 2 4 = 2 2 2 2 = 16 ( 2) 4 = ( 2)( 2)( 2)( 2) = 16 When no exponent is shown, the exponent is 1 𝑥𝑥 = 𝑥𝑥 1 3 4 = 81 ( 7) 2 = 49 7 2 = 49 𝑥𝑥 0 + 6 0 = 2 Product Rule When you multiply 2 monomials, multiply the coefficients and add the exponents 2 𝑥𝑥 5 7 𝑥𝑥 3 = 14x 8 4 𝑛𝑛 ∙ 3 𝑛𝑛 4 = 12n 5 (4x 3 )(5x 5 )(2x 2 ) = 40x 10 ► If the monomial includes more than 1 variable, work with each variable separately ( 3x 3 y 2 )( 5x 4 y 6 ) = 15x 7 y 8 Power of a Product Rule When the base is already an exponential expression, you raise the coefficient to the given power and multiply the exponents. Work with each variable separately. (2 𝑥𝑥 7 ) 3 = (2x 7 )(2x 7 )(2x 7 ) = 8x 21 (5 𝑥𝑥 3 𝑦𝑦 2 𝑧𝑧 ) 3 = 125x 9 y 6 z 3 ( 7 𝑎𝑎 3 𝑏𝑏 3 ) 2 = 49a 6 b 6
Page 2 of 10 10.1 -Exponents Power of a Quotient Rule Same as power of a product 2𝑥𝑥 2 𝑦𝑦 3 = 2x 2 y � � 2x 2 y � � 2x 2 y = 8x 6 y 3 𝑎𝑎 2 𝑏𝑏 𝑐𝑐 3 5 = a 10 b 5 c 15 −2𝑥𝑥 3 𝑦𝑦 7 𝑧𝑧 4 = 16x 12 y 28 z 4 Quotient Rule Divide the coefficients. Keep the base and subtract the exponents. 12𝑦𝑦 6 2𝑦𝑦 3 = 12∙y∙y∙y∙y∙y∙y 2∙y∙y∙y = 6 y y y = 6y 3 20𝑥𝑥 7 5𝑥𝑥 2 = 4x 7−2 = 4x 5 𝑚𝑚 3 𝑚𝑚 3 = m 3−3 = m 0 = 1 𝑥𝑥 11 𝑥𝑥 3 = x 8 6 11 6 3 = 6 8 𝑥𝑥 12 𝑦𝑦 5 𝑥𝑥 8 𝑦𝑦 4 = x 4 y 8𝑎𝑎 3 𝑏𝑏 8 𝑐𝑐 3 18𝑎𝑎𝑐𝑐 3 = 4a 2 b 8 9 Practice Set 10.1 Exponents Summary Handout
Page 3 of 10 10.2 - Negative Exponents & Scientific Notation Negative Exponents 𝑥𝑥 2 𝑥𝑥 5 = x∙x x∙x∙x∙x∙x = 1 x 3 or x 2−5 = x −3 To simplify a base with a negative exponent, take the reciprocal of the base and make the exponent positive Note: You are only taking the reciprocal of the base, not the coefficient 8 𝑚𝑚 −5 = 8 m 5 3 −2 = 1 3 2 7 𝑥𝑥 −4 = 7x 4 𝑥𝑥 −4 𝑦𝑦 −3 = y 3 x 4 𝑥𝑥 𝑥𝑥 −7 = x x 7 = x 8 𝑥𝑥 𝑥𝑥 −7 = x 1− ( −7 ) = x 8 You might say that if a base has a negative exponent, wherever it is, it wants to be someplace else. If the base is on top of a fraction, it wants to be on the bottom. If a base is on the bottom of a fraction, it wants to be on the top. Review 𝑥𝑥 5 ∙ 𝑥𝑥 3 = 𝒙𝒙 𝟖𝟖 When multiplying add exponents ( 𝑥𝑥 5 ) 3 = 𝒙𝒙 𝟏𝟏𝟏𝟏 Exponents raised to another power are multiplied 𝑥𝑥 5 𝑥𝑥 3 = 𝒙𝒙 𝟐𝟐 When dividing, subtract exponents 5 𝑥𝑥 −3 = 𝟏𝟏 𝒙𝒙 𝟑𝟑 Take the reciprocal of bases with negative exponents 𝑥𝑥 0 = 𝟏𝟏 Anything with an exponent of 0 is 1