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Isolating a variable in an equation is a fundamental step in solving equations, as it allows us to express one variable solely in terms of others. Let's break down how to isolate a variable effectively:

1. **Identify the variable to isolate:** Start by determining which variable you're trying to make the subject of the equation. In our example, this variable is \( t \).
2. **Rearrange the equation:** Use basic arithmetic operations to manipulate the equation. Aim to have the variable on one side of the equation and the other terms on the opposite side. In our exercise, we initially have \( s = \frac{1}{2} g t^2 \). To start isolating \( t^2 \), multiply both sides by 2 to remove the fraction, giving us \( 2s = g t^2 \).

3. **Simplify carefully:** As you rearrange, ensure every step is mathematically sound to maintain the equality of the equation. Careful isolation paves the way for further manipulation.

Manipulating a formula involves using arithmetic to transform a given mathematical expression. Here’s a closer look at how we do it:

- **Operations on both sides:** After isolating \( t^2 \) in the equation \( 2s = g t^2 \), the next step involves dividing both sides by \( g \). This operation further simplifies and isolates the term of interest, giving us \( t^2 = \frac{2s}{g} \). - **Maintaining balance:** When adjusting a formula, any operation performed on one side must also be applied to the other. This keeps the original equivalency of the equation intact and leads to accurate results.

- **Achieving a specific form:** These manipulations help express complex relationships in terms of one variable, as seen when we manipulate the formula to ultimately solve for \( t \) in terms of \( s \) and \( g \).

Remember, practice makes perfect. The more formulas you manipulate, the more intuitive these steps become.

Taking the square root is a crucial step in solving equations where a variable is squared. It can transform a squared variable into its linear form, simplifying the expression:

- **Understanding square roots:** A square root finds a number which, when multiplied by itself, gives the original value. Mathematically, \( \sqrt{x} \) is a number \( y \) such that \( y^2 = x \).

- **Applying square roots wisely:** In our example, from the equation \( t^2 = \frac{2s}{g} \), we extracted \( t \) by taking the square root of both sides. This gives us \( t = \sqrt{\frac{2s}{g}} \).

- **Considering positive and negative roots:** Normally when solving equations, square roots yield two possibilities: one positive and one negative solution. However, in realistic scenarios like solving for time \( t \), we use only the positive root since time cannot be negative.

Taking the square root correctly can untangle a complex algebraic expression, offering a straightforward path to the solution.