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 = x34 = x12 Solution b) (42)8 = 428 = 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) (x4)3 c) (3a2b4)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 –