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In algebra, isolating a variable means rearranging an equation so that the variable you're solving for is by itself on one side of the equation. This makes it easier to solve for that particular variable.
Start by identifying the variable you need to solve for. In the given problem, our target variable is the letter \( w \).
This step is important because it sets the stage for the rest of your solution steps.

To isolate a variable, here are some steps to follow:

• Move terms that do not contain the variable to the other side of the equation.
• Use basic arithmetic operations like addition, subtraction, multiplication, or division to isolate the variable.
• Keep the equation balanced by performing the same operations on both sides.

The main goal is to get the variable by itself, making it easier to determine its value.

Solving equations is a critical skill in algebra that involves finding the value of a variable that makes the equation true. In the given exercise, we're looking to solve for \( w \) in the equation \( g w^{2}=2 r \).
Here's a general approach to solve an equation:

• Identify the target variable (which we did).
• Isolate the term containing the target variable (this results in our simplified form).
• Solve for the variable.

In our problem, after isolating the variable \( w \) in the equation, we reach an intermediate step:

• We divided both sides by \( g \) to isolate the term containing \( w \):
• \( w^{2} = \frac{2r}{g} \).

Balancing the equation is key to solving it correctly. Always remember to apply operations uniformly across the equation.

The square root operation is a method used to find a number which, when multiplied by itself, gives the original number.
In our equation \( w^{2} = \frac{2r}{g} \), we want to solve for \( w \). The final step involves taking the square root of both sides to find \( w \).

When you take the square root of both sides, it looks like this:

• \( w = \sqrt{\frac{2r}{g}} \).

The square root removes the exponent on \( w^{2} \), giving us the value of \( w \).

Taking square roots is a useful tool in algebra that helps simplify equations, especially when dealing with squared terms. Remember, it's essential to consider both the positive and negative roots when dealing with square roots, but in the context of this problem, we'll assume the positive root for simplicity.