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

2024
Views:

393

Pages:

3
Author:

Nevaeh Bauer

Question 1 To rationalize: Each numerator. Assume that variables represent positive real numbers. Given: An expression: √9 / 7 Answer Concept used: In order to rationalize the numerator of an expression, we multiply in the numerator and denominator with such a term/expression such that we don't have any radicals left in the numerator upon simplifying. Calculation: Let us rationalize the numerator as shown below: √9 / 7 (Given expression) (√9 / 7) * (√3/√3) (multiplying fraction by √√3) (√9√3) / (7√3) = (√27) / (7√3) = (3√3) / (7√3) (Apply integer√a · √b = √ab) Therefore, the rationalized form of the given expression would be 3√3 / 7√3 Question 2 To calculate: The simplified value of the radical expression √18a ÷ √2a⁴. Answer Given expression is √18a ÷ √2a⁴. Formula used: The product rule for radicals, √ab = √a · √b Here, n is the positive integer and a and b are real number. The relation between radical and rational exponent notation is expressed as, ⁿ√a = a^(1/n). Here, a is the radicand, and n is the index of the radical. Calculation: Consider the given expression, √18a ÷ √2a⁴. Use the product rule for radicals to simplify the expression, √18a ÷ √2a⁴ = √(18a / 2a⁴) = √9a / a⁴ = √(9 * 2 * a) / √(2a²a²) = √9 * √a / √a² * √a² = 3√a / a * a = 3√a / a² Use the relation between radical and rational exponent notation to simplify the expression, √18a ÷ √2a⁴ = (18a)^(1/2) / (2a⁴)^(1/2) = (3² * 2 * a)^(1/2) / (2 * a² * a²)^(1/2) = (3)^(21/2) √a / a^(21/2) * a^(2*1/2) = (3)√a / a * a = 3√a / a² Hence, the simplified value of the provided expression is 3√a / a² Question 3 Simplify the expression and express the answer using rational exponents. Assume that all letters denote positive numbers. ³√x² Answer Concept used: If a is real number n is a positive integer and ⁿ√a^m is a real number then the rational exponent expression ⁿ√a^m is equivalent to radical expression a^(m/n). The above statement can be express as, ⁿ√a^m = a^(m/n) Calculation: The given expression is ³√x². The property of nth is, ⁿ√a^m = a^(m/n) Substitute 2 for n, 3 for m and x for a in the above equation. ³√x² = (x²)^(1/3) = x^(2/3) Hence, the solution of the expression ³√x² is x^(2/3) Question 4 Combining radicals simplify the expression. Assume that all letters denote positive numbers. √16x + √x⁵ Answer Property of: √ab = √a√b Distributive property: a(b+c) = ab + ac Calculation: Apply exponent rule simplify the expression, √16x + √x⁵ = √(2⁴x) + √(x⁴ * x) = √(2⁴) * √x + √(x⁴) * √x = 2²√x + x²√x Using distributive property. = √x (2² + x²) = √x (4 + x²) Answer: √16x + √x⁵ = √x (4 + x²) Question 5 To evaluate the root √100. Answer Concept used: Find the prime factors of the number inside the factor and then take out a number out side of the root for each pair. Calculation: The prime factorization of 100 is 100 = 2 × 2 × 5 × 5. Thus, we can evaluate the root as shown below. √100 = √(2 × 2 × 5 × 5) = √((2 × 2) × (5 × 5)) (Make pairs of same numbers) = √(2²) × √(5²) = 2 × 5 (Take out one number for each pair out side the root) = 10 Conclusion: The value of √100 is 10.