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

Posted by Larry
on 20240118 11:21:51 