The task: evaluate
{(√50-√49)}^(1/3)
may be accomplished by several methods, inter alia by:
Pure algebra
Calculus
Calculator
Trial and error
Computer programs
Other
Please solve by at least 2 distinct ways and compare results.
Calculator:
Resolves to √2 - 1 (technically, 0.414213562373096)
Algebra:
X = (5√2 - 7)^(1/3)
X^3 = (5√2 - 7)
Let X = a + b√2
X^3 = a^3 + 2√2 + 3ab√2(a + b√2)
= a^3 + 2√2 + 3a^2b√2 + 6ab^2
= a^3 + + 6ab^2 + (2 + 3a^2b)√2
a^3 + 6ab^2 = -7
(2 + 3a^2b) = 5 --> a^2b = 1
b = 1/a^2
a^3 + 6a^-3 = -7
a^6 + 7a^3 + 6 = 0
a^3 = {-1,-6}
a = {-1, -∛6 }
b = {1, 6^(-2/3)}
(a,b) = (-1,1) --> X = √2 - 1 checks
Another algebra check:
X = (5√2 - 7)^(1/3)
X^3 = (5√2 - 7)
X^6 = 50 - 70√2 + 49 = 99 - 70√2
(√2 - 1)^3 = 2√2 - 3*2 + 3*√2 - 1 = 5√2 - 7
(5√2 - 7)^2 = 50 - 70√2 + 49 checks
Program:
def f(x):
return ((x**2 + 1)**.5 - x) ** (1/3)
print(f(7))
OUTPUT: 0.4142135623730956
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Posted by Larry
on 2024-01-18 11:21:51 |
Solution
