University:
Carleton College
Course:
MATH 120 | Calculus 2
Academic year:

2024
Views:

56

Pages:

3
Author:

Nevaeh Bauer

Question 1 How do I solve 7(x - 4)² = 28 in simplest radical form? Answer 1 I recommend dividing both sides by 7. Then you are left with (x - 4)² = 4 May x - 4 be aka y because why not. y² = 4 y = ± 2 (or minus two. Sorry, Descartes!) x - 4 = ± 2 x = 4 ± 2 x = 6 or x = 2 Answer 2 We have 7(x - 4)² = 28, divide both sides by 7 then (x - 4)² = 4 now take square root on both sides then (x - 4) = ± 2 now add 4 on both sides in each equation then x = 6 or x=2 Question 2 How do you simplify the following radical expressions: √(72) / √(36x⁸y²) ? Answer 1 The notations a bit ambiguous, but what I think you're asking about is √(72) / √(36x⁸y²). So, the first thing we want to do is see what we can get outside the radicals. And right off the bat I'm seeing 6² in both the numerator and the denominator. Also, that y² will reduce nicely. As will the x⁸, although not quite as far. So, if you're playing the home game and following along we now have √(6².2) / √(6². (x⁴)². y²). And you may be asking, why write it like that? To make it easier to see what I did next (though if you're comfortable with exponents, you can skip right to this step) So, two things to recall: xᵃ . xᵇ = xᵃᵇ and √z² = z. Applying both of those things is how we 6√2 √2 get to 6x⁴y then to x⁴y And there isn't much more we can do here. Answer 2 How do you simplify the following radical expressions: √(72) / √(36x⁸y²) ? Assuming √(72 / 36x⁸y²) √(72 / 36x⁸y²) = √(18 / 9x⁸y²) Note that √y² = Absolute value(y) = √(9 * 2) / √(9x⁸y²) = 3√2 / 3x⁴y = √2 / x⁴y Note that √18 = √(92) = 3√2 For the case where all variable values represent non-negative quantities, we can write √2 / x⁴y Question 3 Give two mixed radicals that are equivalent to √500. Which answer is in simplest radical form? How do you know? Answer 1 Step 1 Think of the 'factorisation' 500 = 2² x 5³ The rules for non-negative numbers are as follows: √a = a^(1/2) (ab)^c = a^cb^c √(ab) = √a * √b Thus: √500 = √(2² * 5³)= √(2² * 5² * 5) = 25√5 = 10√5 I think it is obvious which is in lowest form. Answer 2 Step 1 Simplifying radicals, in this case a square root, is a matter of finding the largest square root that will divide into √500 In the case of √500, I start with the perfect square greater then 1 and try to find the largest perfect square {1, 4, 9, 16, 25, 36 ...} that divides 500. Now 4 works (4 × 125) but is it the largest. How about 25? It works: 500 = 25 × 20 But 20 still divides by 4, which is a perfect square. By now you may see that 100 is the largest perfect square that divides 500. 500 = 100 × 5 Thus, using 100, we have √500 = √100 × √5 This uses the property that the square root of a product is equal to the product of the square roots. Thus √500 = √100 × √5 = 10√5 This is considered fully simplified. There are no other perfect squares, other than 1, that can be factored out of 10√5 Question 4 When 10 times the square root of 48 is written in simplest radical form, the result is k times the square root of 3. What is the value of k? Answer 1 Step 1 10 × √48 = 10 × √(16 × 3) = 10 × 4 × √3 = 40 × √3 k√3 = 40 × √3 → k = 40 Answer 2 Step 1 This is a convoluted way of asking to simplify the expression 10 * √48 / √3 = 10√(48 / 3) = 10√16 = 10*4 = 40