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

2024
Views:

189

Pages:

5
Author:

Cristian Santos

Question 1 How do you find the square root of 13? Answer 1 The square root of 13 ( 13 13 ) is irrational and approximately lies between 9 = 3 9 =3 and 16 = 4 16 =4. Using Linear Interpolation: 13 ≈ 3.6 (an approximation) . 13 ≈3.6(an approximation). Newton-Raphson Method: Start with 𝑥 0 = 3.6 x 0 =3.6. Iterate using the formula: 𝑥 𝑛 + 1 = 𝑥 𝑛 2 + 𝑛 2 𝑥 𝑛 . x n+1 = 2x n x n 2 +n . First iteration: 𝑥 1 = 3. 6 2 + 13 2 ⋅ 3.6 = 12.96 + 13 7.2 ≈ 3.605. x 1 = 2⋅3.6 3.6 2 +13 = 7.2 12.96+13 ≈3.605. Second iteration gives a more accurate value: 𝑥 2 = 3.60555127547. x 2 =3.60555127547. Using a Calculator: 13 ≈ 3.60555127546. 13 ≈3.60555127546. Answer 2 Using Continued Fractions: Start with an approximation: 13 ≈ 18 5 . 13 ≈ 5 18 . The generalized continued fraction form is: 13 = 3 + 1 5 + 1 5 + 1 5 + … . 13 =3+ 5+ 5+ 5+… 1 1 1 . Simplifying further, we can truncate the fraction to find: 13 ≈ 649 180 ≈ 3.605. 13 ≈ 180 649 ≈3.605. Question 2 How do you simplify the square root of 20? Answer 1 Prime factorize 20: 20 = 2 ⋅ 2 ⋅ 5 = 2 5 . 20 = 2⋅2⋅5 =2 5 . Answer 2 Rewrite 20 = 4 ⋅ 5 20=4⋅5: 20 = 4 ⋅ 5 = 2 5 . 20 = 4⋅5 =2 5 . Decimal form: 2 5 ≈ 4.472135. 2 5 ≈4.472135. Question 3 What is the square root of 216? Answer 1 Factorize 216: 216 = 2 3 ⋅ 3 3 . 216=2 3 ⋅3 3 . Simplify: 216 = 2 3 ⋅ 3 3 = 2 ⋅ 3 ⋅ 6 = 6 6 . 216 = 2 3 ⋅3 3 =2⋅3⋅ 6 =6 6 . Answer 2 Greatest Perfect Square Method: The largest perfect square factor of 216 is 36 36: 216 = 36 ⋅ 6. 216=36⋅6. Simplify: 216 = 36 ⋅ 6 = 6 6 . 216 = 36 ⋅ 6 =6 6 . Question 4 How do you find the square root of 25? Answer The square root of 25 is: 25 = 5. 25 =5. Both 5 5 and − 5 −5 satisfy 𝑥 2 = 25 x 2 =25. Question 5 How do you find the square root of 81? Answer 1 The square root of 81 is: 81 = 9. 81 =9. Both 9 9 and − 9 −9 satisfy 𝑥 2 = 81 x 2 =81. Answer 2 Factorize 81: 81 = 9 ⋅ 9 or 3 4 . 81=9⋅9or3 4 . Simplify: 81 = 9. 81 =9. Answer 3 Using the Babylonian Method: Guess an initial value 𝑥 1 = 8.3 x 1 =8.3. Use the formula: 𝑥 𝑛 + 1 = 1 2 ( 𝑥 𝑛 + 𝑁 𝑥 𝑛 ) , 𝑁 = 81. x n+1 = 2 1 (x n + x n N ),N=81. Iterate until you reach convergence at 𝑥 = 9 x=9.