a and b are positive numbers that satisfy the equation 1/a−1b=1/a+b. Determine the value of a^6/b^6+b^6/a^6.
Solving you would get
b^2 - a^2 = ab -> b/a + a/b = 1,
now let a/b =x,
so x + 1/x = 1 -> x^2-x+1 =0
Solve for x = [1+ root(5)]/2 = a/b
To determine the value of a^6/b^6 + b^6/a^6, we need to find the values of a and b that satisfy the given equation. Let's solve the equation first.
The given equation is: 1/a - 1/b = 1/a + b
To simplify the equation, we can add (1/a) to both sides:
1/a - 1/b + 1/a = 1/a + b + 1/a
Combining like terms, we get:
2/a - 1/b = 2/a + b
Now, let's subtract (2/a) from both sides:
2/a - 2/a - 1/b = b
Simplifying further, we have:
-1/b = b
Multiplying both sides by -b, we get:
1 = -b^2
Since 'b' is a positive number, this equation has no real solutions. Therefore, there are no values of 'a' and 'b' that satisfy the given equation.
As a result, we cannot determine the value of a^6/b^6 + b^6/a^6 as there are no solutions to the equation.