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

2024
Views:

417

Pages:

3
Author:

Nevaeh Bauer

Question 1 We need to calculate: the simplified form of (3√a³(-5√a³)) Answer Given: (3√a³(-5√a³)) Used formula: For a real number a when m is even, √a^m = |a| Calculation: (3√a³(-5√a³)) The radicals are like radicals and have the same radicand & same index. Regrouping the factors and applying the product property of radicals: (3√a³(-5√a³)) = 3 × (-5)√a³ * √a³ Apply property of exponents: 3 × (-5)√a³ * √a³ = -15√a⁶ -15√a⁶ = -15a³ * a³ = -15 a^6 Rewrite the expression with rational exponent: -15√a⁶ = -15(a⁶)^(1/2) = -15(a⁶)^(1/2) = -15 a^(6*1/2) = -15a³ Question 2 Simplify expression and express your answer using rational indicators. Suppose all letters represent positive numbers. 1 / ³√a^m Answer Step 1 Concept used: If a - a real number, n - positive integer ⁿ√a^m = a^(m/n) The above statement can be expressed as, 1 / ⁿ√a^m = a^(-m/n) Step 2 The property of n the root after combine the 1/√a^n = a^(-n) and √a^m = a^(m/n) is, 1 / ³√a^m = a^(-m/3) Substitute 5 for n, 3 for m and x for a in the above equation. 1 / ³√x³ = x^(-3/5) = (x³)^(-1/5) = x^(-3/5) Thus, the solution of the expression 1 / ³√x³ is x^(-3/5) Question 3 The simplified value of the radical expression √(½) Answer Step 1 Formula: (ⁿ√a)^m = ⁿ√a^m a) √ab = √a * √b b) ⁿ√a/b = ⁿ√a/ ⁿ√b Step 2 a) √(1/2) = √1 / √2 * √2/√2 = √2 / 2 b) ³√(4/27) = ³√4 / ³√27 = ³√2^2 / 3 = ³√4 / 3 = 2 / ³√9 Question 4 Find the radical form of the expression 25x^(½). Answer Given: 25x^(½) Concept used: x^(½) = √x Calculation: 25x^(½) = 25√x Question 5 To write the expression ³√256a³ in exponential form. Answer Given: ³√256a³ Concept used: x^(½) = √x Calculation: ³√256a³ = (256a³)^(1/3) = (256)^(1/3) * (a³)^(1/3) = (4^4)^(1/3) * a = 4^(4/3) * a = 4*(4^(1/3))* a = 4a³√4