How do I simplify Y raised to A Fraction?

Solving equations containing Fractions as exponents to variables is similar to solving Radicals with variables. We can simplify the fractions in exponents by application of logs or by using some mathematical operations. Let us see how do i simplify y raised to a fraction with the help of an examples as follows:

Example: Suppose we have an equation in y as: ((y) ^1/2 + 2)) / ((y) ^1/2 + 5) = 1 /2. Then find the value of 'y'?

Solution: In the given equation we can solve for 'y' by 1st cross multiplying the numerators and denominators on both sides as follows:

((y) ^1/2 + 2)) / ((y) ^1/2 + 5) = 1 /2,

2 ((y) ^1/2 + 2)) = ((y) ^1/2 + 5),

Squaring both sides we get:

Or 4 ((y) ^1/2 + 2)) ² = ((y) ^1/2 + 5)^{2 _,}

Or 4 y + 16 + 16 y^1/2 = y + 25 + 10 ^{y1/2 _,}

Or 3 y – 9 = - 6 √y,

Or y – 3 = -2 √y,

Squaring again we get:

(y – 3) ² = (-2 √y)^2_,

Or y² + 9 – 6 y = 4 y,

Or y² – 10 y + 9 = 0

Finally we get two values for y as the roots of the above Quadratic Equation as: a² – 10 a + 9 = 0

or y ( y – 9) – (y – 9) = 0,

or y = 1, 9

We can confirm the values of y by substituting them in the original equation as follows:

For y = 1

(√y) + 2)) / (√y + 5) = 1 /2,

3 /6 = 1 /2 = RHS

And for y = 9,

(√9) + 2)) / (√9 + 5) = 1 /2,

5 /8 ≠½,

Thus only y = 1 is true.