91Ó°ÊÓ

When tackling any equation, your goal is to find the value of the unknown variable that makes the equation true. This process generally involves several steps: understanding the given equation, manipulating it to isolate the variable, and solving for the variable. In our example, the equation is \(\root[3]{x-2} = 3\). Here's how you solve it step-by-step:

• Understand the equation: Recognize that \(\root[3]{x-2}\) represents the cube root of \(x-2\)
• Eliminate the root: Cube both sides of the equation
• Isolate the variable: Solve for \(x\) after removing the root

This systematic approach will help you solve complex equations with ease.

Roots, such as square roots or cube roots, often complicate equations. To simplify these, you need to 'eliminate' or remove the root. In our example, we dealt with a cube root \(\root[3]{x-2} = 3\), and here's how you can eliminate it:

To remove a cube root, you cube both sides of the equation. This is because cubing and taking the cube root are inverse operations. Cubing the left side, \(\left( \root[3]{x-2} \right)^3\), cancels the cube root, leaving us with \(x-2\). On the right side, we compute \(3^3\), which equals 27. Thus, the equation \(\root[3]{x-2} = 3\) simplifies to \(x-2 = 27\).

Removing roots can simplify your equation drastically, making it easier to solve for the variable.

Once you've simplified your equation by eliminating roots or other complex parts, the next step is to isolate the variable. This means you need to get the variable by itself on one side of the equation. In our example, after eliminating the cube root, we were left with \(x-2 = 27\).

To isolate \(x\), you need to perform operations that will leave \(x\) alone on one side. Add 2 to both sides:

• Add 2 to \(x-2 \)
• Add 2 to 27

So, \(x-2 + 2 = 27 + 2\), which simplifies to \ x = 29 \. This process of isolation helps you find the value of the variable, completing your solution. Always check your solution by plugging the value back into the original equation.