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MATH 5: HOMEWORK 7 MARCH 3, 2013 Square Root Question: What is the square of a number, say 2? Answer: The square of 2 is 4 because 2 × 2 = 4. Exercise: What is the square of 5? √ The square root of a number, say 4 is 2 because 2 × 2 = 4. We write it as: 4 = 2. Note that also (−2) × (−2) = 4, but for the moment we will be considering only positive solutions. √ In general the square root of a number a2 is a; we can write it as a2 = a. To easily find the squre root of a number, write the prime factorization for the number and group them as (...)2 . For example: √ 484 = 2 × 2 × 11 × 11 = 22 × 112 = (2 × 11)2 So 484 = 2 × 11 = 22. The square of a Sum We learned how to compute using the distributive property: (a + b)(x + y) = a(x + y) + b(x + y) = ax + ay + bx + by What happens when we want to multiply the same sum? (a + b)2 = (a + b)(a + b) = a(a + b) + b(a + b) = aa + ab + ba + bb = a2 + 2ab + b2 (a + b)2 = a2 + 2ab + b2 (1) Example: What is (x + 1)2 ? We can use the formula and save time! (x + 1)2 = x2 + 2x + 1 Exercise: Compute (x + 2)2 = The square of a Difference Similarly when multiplying the same difference with itself: (a − b)2 = (a − b)(a − b) = a(a − b) − b(a − b) = aa − ab − ba + bb = a2 − 2ab + b2 (2) Exercise: Compute (x − 3)2 = (a − b)2 = a2 − 2ab + b2 The difference of Squares (a + b)(a − b) = a(a − b) + b(a − b) = aa − ab + ba − bb = a2 − b2 Let’s write this result starting with the rightmost side: a2 − b2 = (a − b)(a + b) (3) In other words, you can always write a difference of 2 squares as a product of 2 factors. Note that 12 = 1, so x2 − 12 = x2 − 1 = (x − 1)(x + 1). Exercise: Factor 4x2 − 1 = Homework 1. Write the squares of all consecutive numbers from 1 to 20. √ √ √ √ 2. What is 144, 169, 100, 676? 3. Compute: (a)(2x)2 = √ (d) 121 × 169 = (b)(4xy)2 = √ (e) 22 × 34 = (c)(5y 4 )2 = √ (f) 10000 = (Remember the rule (ab)m = am bm when we talked about exponents, you have to raise to the power each factor. For example: (3abc)2 = 32 a2 b2 c2 = 9a2 b2 c2 ) 4. Compute using difference of squares: (a)142 − 42 = (b)262 − 42 = (c)1212 − 192 = 5. What is: (a)(x − 5)2 = (b)(x + 2)2 = (c)y 2 − x2 = 6. Solve the Math Kangaroo problems in the given handout.