University:
Johns Hopkins University
Course:
AS.110.601 | Algebra I
Academic year:

2024
Views:

84

Pages:

5
Author:

Cristian Santos

Question 1 What is the square root of 80? Answer 1 Using square root properties: 80 = 4 × 20 = 4 × 4 × 5 = 4 5 . 80 = 4×20 = 4×4×5 =4 5 . Approximate decimal value: 80 ≈ 8.944. 80 ≈8.944. Answer 2 Prime factorize 80 80: 80 = 2 × 2 × 2 × 2 × 5. 80=2×2×2×2×5. Group the factors in pairs: ( 2 × 2 ) × ( 2 × 2 ) × 5 = 2 2 ⋅ 2 2 ⋅ 5 . (2×2)×(2×2)×5 = 2 2 ⋅2 2 ⋅5 . Apply 𝑎 2 = 𝑎 a 2 =a: 80 = 2 ⋅ 2 5 = 4 5 . 80 =2⋅2 5 =4 5 . Question 2 How do you simplify 1 2 2 1 ? Answer 1 Multiply numerator and denominator by 2 2 : 1 2 × 2 2 = 2 2 ⋅ 2 . 2 1 × 2 2 = 2 ⋅ 2 2 . Simplify the denominator: 2 ⋅ 2 = 2. 2 ⋅ 2 =2. Result: 2 2 . 2 2 . Answer 2 Directly multiply 1 2 2 1 by 2 2 2 2 : 2 2 . 2 2 . Question 3 For each set of numbers, list the appropriate elements. Answer a) Natural Numbers: 21 21 b) Whole Numbers: 21 , 0 21,0 c) Integers: − 6 , 21 , 0 −6,21,0 d) Rational Numbers: − 5 , 21 , 5.62 , 0.4 , 2 3 , 0 , − 7 8 −5,21,5.62,0.4, 3 2 ,0,− 8 7 e) Irrational Numbers: 3 , 𝜋 3 ,π f) Real Numbers: All numbers provided in the set. Question 4 Prove or disprove: If 𝑎 a and 𝑏 b are rational numbers, then 𝑎 𝑏 a b is also rational. Answer 1 Counterexample: Let 𝑎 = − 1 a=−1 and 𝑏 = 1 2 b= 2 1 . 𝑎 𝑏 = ( − 1 ) 1 / 2 = − 1 , a b =(−1) 1/2 = −1 , which is not a rational number. Result: The statement is false. Answer 2 Counterexample: Let 𝑎 = 2 a= 2 and 𝑏 = 1 2 b= 2 1 : 𝑎 𝑏 = ( 2 ) 1 / 2 = 2 . a b =( 2 ) 1/2 = 2 . Since 2 2 is irrational, 2 2 is also irrational. Answer 3 Let 𝑎 = 2 a=2 and 𝑏 = 1 2 b= 2 1 : 𝑎 𝑏 = 2 1 / 2 = 2 , a b =2 1/2 = 2 , which is irrational. Question 5 Explain why the sum, difference, and product of two rational numbers are rational. Are the sum and product of two irrational numbers always irrational? Answer Step 1: The sum, difference, and product of two rational numbers 𝑎 𝑏 b a and 𝑐 𝑑 d c are rational because: Sum: 𝑎 𝑏 + 𝑐 𝑑 = 𝑎 𝑑 + 𝑏 𝑐 𝑏 𝑑 . b a + d c = bd ad+bc . Difference: 𝑎 𝑏 − 𝑐 𝑑 = 𝑎 𝑑 − 𝑏 𝑐 𝑏 𝑑 . b a − d c = bd ad−bc . Product: 𝑎 𝑏 ⋅ 𝑐 𝑑 = 𝑎 𝑐 𝑏 𝑑 . b a ⋅ d c = bd ac . All these results are rational since 𝑎 , 𝑏 , 𝑐 , 𝑑 ∈ 𝑍 a,b,c,d∈Z. Step 2: The sum or product of two irrational numbers is not always irrational. Example: Product of 3 3 and 1 3 3 1 : 3 ⋅ 1 3 = 1 ( rational ) . 3 ⋅ 3 1 =1(rational). Sum of 𝜋 π and − 𝜋 −π: 𝜋 + ( − 𝜋 ) = 0 ( rational ) . π+(−π)=0(rational).