Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §1.6 Exponent Properties Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
1.5 Review § Any QUESTIONS About Any QUESTIONS About HomeWork MTH 55 Review § Any QUESTIONS About §1.5 → (Word) Problem Solving Any QUESTIONS About HomeWork §1.5 → HW-01
Exponent PRODUCT Rule For any number a and any positive integers m and n, Exponent Base In other Words: To MULTIPLY powers with the same base, keep the base and ADD the exponents
Quick Test of Product Rule
Example Product Rule Multiply and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a) x3 x5 b) 62 67 63 c) (x + y)6(x + y)9 d) (w3z4)(w3z7)
Example Product Rule Solution a) x3 x5 = x3+5 Adding exponents Solution b) 62 67 63 = 62+7+3 = 612 Solution c) (x + y)6(x + y)9 = (x + y)6+9 = (x + y)15 Solution d) (w3z4)(w3z7) = w3z4w3z7 = w3w3z4z7 = w6z11 Base is x Base is 6 Base is (x + y) TWO Bases: w & z
Exponent QUOTIENT Rule For any nonzero number a and any positive integers m & n for which m > n, In other Words: To DIVIDE powers with the same base, SUBTRACT the exponent of the denominator from the exponent of the numerator
Quick Test of Quotient Rule
Example Quotient Rule Divide and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.) a) b) c) d)
Example Quotient Rule Solution a) Base is x Solution b) Base is 8 Solution c) Base is (6y) Solution d) TWO Bases: r & t
The Exponent Zero For any number a where a ≠ 0 In other Words: Any nonzero number raised to the 0 power is 1 Remember the base can be ANY Number 0.00073, 19.19, −86, 1000000, anything
Example The Exponent Zero Simplify: a) 12450 b) (−3)0 c) (4w)0 d) (−1)80 e) −80 Solutions 12450 = 1 (−3)0 = 1 (4w)0 = 1, for any w 0. (−1)80 = (−1)1 = −1 −80 is read “the opposite of 80” and is equivalent to (−1)80: −80 = (−1)80 = (−1)1 = −1
The POWER Rule For any number a and any whole numbers m and n In other Words: To RAISE a POWER to a POWER, MULTIPLY the exponents and leave the base unchanged
Quick Test of Power Rule
Example Power Rule Simplify: a) (x3)4 b) (42)8 Solution a) (x3)4 = x34 = x12 Solution b) (42)8 = 428 = 416 Base is x Base is 4
Raising a Product to a Power For any numbers a and b and any whole number n, In other Words: To RAISE A PRODUCT to a POWER, RAISE Each Factor to that POWER
Quick Test of Product to Power
Example Product to Power Simplify: a) (3x)4 b) (−2x3)2 c) (a2b3)7(a4b5) Solutions (3x)4 = 34x4 = 81x4 (−2x3)2 = (−2)2(x3)2 = (−1)2(2)2(x3)2 = 4x6 (a2b3)7(a4b5) = (a2)7(b3)7a4b5 = a14b21a4b5 Multiplying exponents = a18b26 Adding exponents
Raising a Quotient to a Power For any real numbers a and b, b ≠ 0, and any whole number n In other Words: To Raise a Quotient to a power, raise BOTH the numerator & denominator to the power
Quick Test of Quotient to Power
Example Quotient to a Power Simplify: a) b) c) Solution a) Solution b) Solution c)
Negative Exponents Integers as Negative Exponents
Negative Exponents For any real number a that is nonzero and any integer n The numbers a−n and an are thus RECIPROCALS of each other
Example Negative Exponents Express using POSITIVE exponents, and, if possible, simplify. a) m–5 b) 5–2 c) (−4)−2 d) xy–1 SOLUTION a) m–5 = b) 5–2 =
Example Negative Exponents Express using POSITIVE exponents, and, if possible, simplify. a) m–5 b) 5–2 c) (−4)−2 d) xy−1 SOLUTION c) (−4)−2 = d) xy–1 = Remember PEMDAS
More Examples Simplify. Do NOT use NEGATIVE exponents in the answer. a) b) (x4)3 c) (3a2b4)3 d) e) f) Solution a)
More Examples Solution b) (x−4)−3 = x(−4)(−3) = x12 c) (3a2b−4)3 = 33(a2)3(b−4)3 = 27 a6b−12 = d) e) f)
Factors & Negative Exponents For any nonzero real numbers a and b and any integers m and n A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed
Examples Flippers Simplify SOLUTION We can move the negative factors to the other side of the fraction bar if we change the sign of each exponent.
Reciprocals & Negative Exponents For any nonzero real numbers a and b and any integer n Need to make a TEST for this Any base to a power is equal to the reciprocal of the base raised to the opposite power
Examples Flippers Simplify SOLUTION
Summary – Exponent Properties 1 as an exponent a1 = a 0 as an exponent a0 = 1 Negative Exponents (flippers) The Product Rule The Quotient Rule The Power Rule (am)n = amn The Product to a Power Rule (ab)n = anbn The Quotient to a Power Rule This summary assumes that no denominators are 0 and that 00 is not considered. For any integers m and n
WhiteBoard Work Problems From §1.6 Exercise Set 14, 24, 52, 70, 84, 92, 112, 130 Base & Exponent → Which is Which?
Astronomical Unit (AU) All Done for Today Astronomical Unit (AU)
Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –