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When solving quadratic equations, the first step is usually to isolate the variable term. This means arranging the equation so that the term with the variable squared is by itself on one side of the equality sign. To do this, we often need to perform operations that maintain the balance of the equation, like adding or subtracting the same number from both sides, or multiplying or dividing by a non-zero constant.

In the exercise with the equation \(9 w^{2} - 64 = 0\), isolating the variable term \(w^{2}\) involves adding 64 to both sides to cancel out the -64. This step is crucial because it simplifies the equation to a form where the square root method can be applied directly. Remember, keeping the balance of the equation is essential whenever we isolate any variable term.

The square root method is particularly handy when we have a single squared variable term equals to a number. After isolating this term, as we did in the previous step with \(w^{2} = 64/9\), you can apply square roots on both sides to solve for \(w\). Doing so allows us to reverse the operation of squaring the variable.

It's important to remember to include both the positive and negative square roots, so we use \(\pm\) to indicate that both roots are solutions to the quadratic equation. In our exercise, this gives us \(w = \pm \sqrt{64/9}\), which leads us to two possible values for \(w\). The square root method is straightforward but always check for any constraints in the original problem that might prefer one solution over the other.

After applying the square root method, simplifying the obtained algebraic expression is the next step. Simplification makes the solution clear and easy to understand. We seek to rewrite the expression in the simplest form possible.

For instance, with \(w = \pm \sqrt{64/9}\), simplifying the square root of a fraction involves taking the square root of the numerator and the denominator separately. Since the square root of 64 is 8, and the square root of 9 is 3, the simplified expression is \(w = \pm 8/3\). Such simplification not only makes the results neater but often is necessary for further calculations, such as graphing the function or using the values of \(w\) in subsequent arithmetic operations.