x^1/2-6=x^1/4
please explain steps to solving!
√x - 6 = x^(1/4)
let √x = t
then we have
t - 6 = √t
square both sides
t^2 - 12t + 36 = t
t^2 - 13t + 36 = 0
(t-9)(t-3) = 0
t = 9 or t = 3
if t = 9, then x = 81
if t = 3, then x = 9
since we squared our equation, all answers must be verifies
if x = 81
LS = √81 - 6 = 9-6 = 3
RS = 81^(1/4) = 3
so x = 81 is valid
if x = 9
LS = √9 - 6 = 3-6 = -3
RS = 9^1/4) ≠ -3
so the only solution is x = 81
if we let t = ∜x, we have
t^2-6 = t
t^2-t-6 = 0
(t-3)(t+2) = 0
t = 3 or -2
so, x = 81 or 16
But, since ∜x is positive, only x=81 satisfies the original equation.
So, the extraneous solution can arise even when the original equation is not squared. Trying to use x=16 gives us
√16 - 6 = ∜16
But
√16 - 6 = 4-6 = -2
and -2 is not ∜16
To solve the equation x^(1/2) - 6 = x^(1/4), we can follow these steps:
1. Start by isolating one of the terms with a rational exponent. In this case, let's isolate the term x^(1/4). We can do this by subtracting x^(1/4) from both sides of the equation:
x^(1/2) - x^(1/4) - 6 = 0
2. Now, let's simplify the left side of the equation. To simplify a term with a fractional exponent, we can use the property of exponents that states: (a^m)^n = a^(m*n). Applying this property, we get:
x^(1/2) * x^(-1/4) - 6 = 0
3. Next, let's simplify further. Remember that x^(-n) is equal to 1/x^n. By applying this rule, we can rewrite the equation as:
x^(1/2) / x^(1/4) - 6 = 0
(x^(1/2))/(x^(1/4)) - 6 = 0
4. To combine the fractions, we need a common denominator. In this case, the least common denominator is x^(1/4). Let's adjust the fractions:
(x^(1/2) - 6x^(1/4))/(x^(1/4)) = 0
5. Now, multiply both sides of the equation by x^(1/4) to eliminate the denominator:
(x^(1/2) - 6x^(1/4)) = 0 * x^(1/4)
x^(1/2) - 6x^(1/4) = 0
6. To solve for x, we can treat it as a variable raised to a power. Let's substitute y = x^(1/4), so the equation becomes:
y^2 - 6y = 0
7. This equation is now quadratic in form. Factoring out the common term y, we get:
y(y - 6) = 0
8. Setting each factor equal to zero, we have two possibilities:
y = 0 or y - 6 = 0
9. Solving these equations gives:
y = 0 or y = 6
10. Recall that y = x^(1/4), hence:
x^(1/4) = 0 or x^(1/4) = 6
11. To eliminate the fractional exponent, raise both sides of each equation to the power of 4:
(x^(1/4))^4 = 0^4 or (x^(1/4))^4 = 6^4
x = 0 or x = 1296
So, the solutions to the equation x^(1/2) - 6 = x^(1/4) are x = 0 and x = 1296.