91Ó°ÊÓ

Before solving any radical equation, the first step is to isolate the radical term. By doing so, we focus solely on the part of the equation that contains the radical expression. In the given exercise, the equation is:
$$2 y^{1 / 2}-13=7$$
We add 13 to both sides to move the constant term to the other side:
$$2y^{1/2}-13+13 = 7+13$$
This gives us:
$$2y^{1/2} = 20$$
Now, we divide both sides by 2 to isolate the radical term:
$$\frac{2y^{1/2}}{2} = \frac{20}{2}$$
This simplifies to:
$$y^{1/2} = 10$$
Remember, isolating the radical term involves getting the radical expression by itself on one side of the equation. This sets the stage for the next steps in solving the equation.

Once we have isolated the radical term, the next step is to remove the radical. We do this by squaring both sides of the equation. In our isolated radical equation:
$$y^{1/2}=10$$
Note that the radical expression is the square root, denoted as $$y^{1/2}$$. To eliminate the radical, square both sides:
$$\big(y^{1/2}\big)^2 = 10^2$$
Squaring $$y^{1/2}$$ returns $$y$$, and squaring 10 yields:
$$y=100$$
This step is crucial because it transforms a more complex radical equation into a simpler polynomial equation, making it easier to solve.

After isolating the radical term and squaring both sides, we typically arrive at a simplified equation. Simplifying equations involves performing basic algebraic operations until you find the value of the variable. From our current equation:
$$y=100$$
Here, we already have the solution. Simplifying can involve additional steps in other cases, such as combining like terms or performing inverse operations to isolate the variable.
Key tips for simplifying equations:

• Always perform the same operation on both sides
• Check each step to ensure accuracy
• Substitute the solution back into the original equation to verify correctness

Simplifying correctly ensures that we derive an accurate and valid solution to the equation.