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

2024
Views:

83

Pages:

3
Author:

Nevaeh Bauer

Question 1 To calculate: To rationalize the numerator of the expression (√x - √5) / (x²-25) Answer Calculation: The numerator of a rational expression can be rationalized by removing the irrational term from it. To remove the irrational term, the numerator and denominator of the expression are multiplied by the conjugate of the numerator. Conjugate of an expression √a + √b is given as √a - √b and vice-versa. So the conjugate of the numerator is given as √x + √5. Multiplying the numerator and the denominator with the conjugate. (√x - √5) / (x²-25) * (√x + √5) / (√x + √5) = (√x - √5) (√x + √5) / (x²-25)(√x + √5) = (√x)² - (√5)² / (x²-25)(√x + √5) = (x - 5) / (x²-25)(√x + √5) (Since square and square root are opposite operations, they cancel out) = (x - 5) / (x - 5)(x + 5)(√x + √5) i.e. (a + b)(a - b) = a² - b² = 1 / (x + 5)(√x + √5) Question 2 To calculate: To rationalize the denominator of the expression 7 / √15 Answer Calculation: The denominator of a rational expression can be rationalized by removing the irrational term from it. To remove the irration term, the numerator and denominator of the expression are multiplied by the conjugate of the denominator. Conjugate of an expression √a + √b is given as √a - √b and vice-versa. This can be written as: 7 / (√15 + 0) So the conjugate of the denominator is given as √15 - 0 = √15 Multiplying the numerator and the denominator with the conjugate, 7 / √15 * (√15 / √15) = 7 √15 / √15 * √15 = 7√15 / (√15)² = 7√15 / 15 (since square and square root are opposite operations, they cancel out). The value of 7/√15 = 7√15 / 15 Question 3 The value of ³√64. Answer Given: ³√64 Concept used: ⁿ√{a} = a^(1/n) Calculation: ³√64 = 64^(1/3) And 64 is the cube of 4. So, we can write it as ³√64 = (4³)^(1/3) Since there are two exponents, so we multiply them. And that would result in 1. Therefore ³√64 = 4 Question 4 To simplify the expression 25^(3/2) Answer Calculator: According to the power rule (bⁿ)ᵐ = bⁿᵐ Given expression is 25^(3/2) This can be written as 25^(3/2) = (5 * 5)^(3/2) = (5²)^(3/2) Applying the power rule, 25^(3/2) = (5)^(2(3/2)) 25^(3/2) = 5³ 25^(3/2) = 125. Question 5 Write each expression using rational exponents in radical form 1.) 64^(2/3) 2.) 49^(3/2) 3.) (-243)^(4/5) 4.) 625^(3/4) 5.) (-512)^(2/3) Answer Apply the rule x^(m/n) = ⁿ√x^m to rewrite the exponentiation as a radical. 64^(2/3) = ³√64² Apply the rule x^(m/n) = ⁿ√x^m to rewrite the exponentiation as a radical. 49^(3/2) = √49³ Apply the rule x^(m/n) = ⁿ√x^m to rewrite the exponentiation as a radical. 625^(3/4) = ⁴√625³ (-512)^(2/3) = ³√(-512)² Rewrite 512 as 8³ ³√(-8³)² Cancel the common factor of 3 (-8)² = 64 5) (-243)^(4/5) = ⁵√(-243)⁴ =(-3)⁴ = 81