Definitions Exponents and Order of Operations Objective: To use exponents, the order of operations, and grouping symbols. variable – a symbol, usually a letter, that represents one or more numbers algebraic l b i expression i – a math th phrase h with numbers, variables, and operation symbols Grouping Symbols Definitions & Operation Symbols Addition – sum, more than, increase Subtraction – difference, less than, decrease Multiplication – product, times, of Division – quotient, per ( ) parentheses [ ] brackets { } b braces fraction bar a+b a-b a●b, (a)(b), a(b), (a)b, ab a÷b, a/b, a , b√a b Powers power -- an expression like x2, where x is the base and 2 is the exponent squared d – to t th the 2nd power cubed – to the 3rd power Try these. Expand and multiply. a) 34 b) 82 c) 105 d) 193 1 SOLUTIONS Expand and multiply. a) 34 3(3)(3)(3) = 81 b) 82 8(8)=64 8(8) 64 105 10(10)(10)(10)(10) =100,000 d) 193 (19)(19)(19) 6859 c) Order of Operations 1. 2. 3. 4. Examples P.E.M.D.A.S. 1. 2. 3. 4. Please Excuse My Dear Aunt Sally! Simplify using the Order of Operations. e) 6 – 10 ÷ 5 f) 3 · 6 – 42 ÷ 2 P – parentheses th ((any grouping i symbols) b l ) E – exponents M/D – multiply/divide A/S – add/subtract Examples – Solutions Simplify using the Order of Operations. e) 6 – 10 ÷ 5 f) 3 · 6 – 42 ÷ 2 6–2 3 · 6 – 16 ÷ 2 4 18 – 8 10 Grouping symbols (parentheses, brackets, fraction bar) E Exponents t Multiply and divide in order from left to right. Add and subtract in order from left to right. More Examples Simplify using the Order of Operations. g) 4 · 7 + 4 ÷ 22 h) 53 + 90 ÷ 10 2 Simplify using the Order of Operations. More Examples Simplify using the Order of Operations. g) 4 · 7 + 4 ÷ 22 h) 53 + 90 ÷ 10 4·7+4÷4 125 + 90 ÷ 10 28 + 1 125 + 9 29 134 Simplify using the Order of Operations. SOLUTIONS i) 12 + 4(2 + 3) 2 i) 12 + 4i(2 + 3) 2 Simplify using the Order of Operations. j) 8 − 2[(5 − 3) 2 − 1] 12 + 4(5) 2 8 − 2[(2) 2 − 1] 12 + 4(25) 12 + 100 8 − 2[4 − 1] 8 − 2[3] k) 5 − [3i(4 − 2) 2 ] + (3i5) l) 14 ÷ 7i[12 ÷ (4 − 2) 2 i5 − 3] 8−6 2 112 Simplify using the Order of Operations -- SOLUTIONS k) 5 − [3(4 − 2) 2 ] + (3i5) 5 − [3(2) ] + (15) 2 5 − [3(4)] + 15 5 − [12] + 15 − 7 + 15 8 j) 8 − 2[(5 − 3) 2 − 1] Examples l) 14 ÷ 7[12 ÷ (4 − 2) 2 i5 − 3] 14 ÷ 7[12 ÷ (4 − 2) i5 − 3] 2 14 ÷ 7[12 ÷ (2) 2 i5 − 3] 14 ÷ 7[12 ÷ 4i5 − 3] 14 ÷ 7[3i5 − 3] 14 ÷ 7[15 − 3] 14 ÷ 7[12] 2[12] 24 ⎛3⎞ m) ⎜ ⎟ ⎝4⎠ ⎛ 32 ⎞ =⎜ 2⎟ ⎝4 ⎠ 9 = 16 2 ⎛1⎞ n) ⎜ ⎟ ⎝5⎠ 3 ⎛ 13 ⎞ =⎜ 3⎟ ⎝ 5 ⎠ 1 = 125 ⎛1⎞ o) ⎜ ⎟ ⎝2⎠ 4 ⎛2⎞ i⎜ ⎟ ⎝3⎠ 2 ⎛ 14 ⎞ ⎛ 2 2 ⎞ = ⎜ 4 ⎟⎜ 2 ⎟ ⎝ 2 ⎠⎝ 3 ⎠ ⎛ 1 ⎞⎛ 4 ⎞ = ⎜ ⎟⎜ ⎟ ⎝ 16 ⎠ ⎝ 9 ⎠ 4 1 = = 144 36 3 Examples 5 ⎛ 3 ⎞⎛ 2 ⎞ − ⎜ p) ⎟⎜ ⎟ 9 ⎝ 4 ⎠⎝ 3 ⎠ 5 ⎛ 6 ⎞ = − ⎜ ⎟ 9 ⎝ 12 ⎠ 5 1 − 9 2 10 9 = − 18 18 1 = 18 = 3 ⎛ 2 3 ⎞ i ⎟ ⎜ 4 ⎝ 3 5 ⎠ 3 ⎛ 6 ⎞ = ⎜ ⎟ 4 ⎝ 15 ⎠ Examples q) 3 ⎛ 2 ⎞ ⎜ ⎟ 4 ⎝ 5 ⎠ 6 = 20 3 = 10 = 2 4 ⎛5⎞ r) ⎜ ⎟ ÷ 3 ⎝6⎠ 25 3 = i 36 4 75 = 144 25 = 48 2(7 + 8) + 2 3·5 + 1 2(15) + 2 = 15 + 1 32 = 16 s) =2 4