\(A number x is 3 larger than its reciprocal. What is the value of \left(x-\frac{1}{x}\right)^4?\)
to take the reciprocal of a number you raise it to the power of negative one, or divide 1 by the number (same thing)
so the reciprocal of x would be x^(-1), or just 1/x. we know that a number is 3 larger than its reciprocal, lets call this number x.
so 1/x <-- the reciprocal, plus 3 would be equal to x. we can write this as 1/x + 3 = x
lets add 1/x and 3, and we get (3x+1)/x = x
multiply both sides by x, and we get 3x+1 = x^2
now lets move this into a quadratic that equals zero
x^2-3x-1 = 0
we can use the quadratic formula to find x, and we end up with
x= (3 plusorminus sqrt(13))/2
now plug these values into (x-(1/x))^4
