Evaluate square root of x-2=4

Solution:

Step 1:

Square both sides to eliminate the square root: $(\sqrt{x - 2})^{2} = 4^{2}$.

Step 2:

Expand and simplify both sides of the equation.

Step 2.1:

Express $\sqrt{x - 2}$ as $(x - 2)^{\frac{1}{2}}$: $((x - 2)^{\frac{1}{2}})^{2} = 4^{2}$.

Step 2.2:

Simplify the left-hand side.

Step 2.2.1:

Evaluate $((x - 2)^{\frac{1}{2}})^{2}$.

Step 2.2.1.1:

Apply exponent multiplication: $((x - 2)^{\frac{1}{2}})^{2}$.

Step 2.2.1.1.1:

Use the power of a power rule: $(x - 2)^{\frac{1}{2} \cdot 2} = 4^{2}$.

Step 2.2.1.1.2:

Reduce the exponents: $(x - 2)^{\frac{1}{\cancel{2}} \cdot \cancel{2}} = 4^{2}$.

Step 2.2.1.1.2.1:

Simplify the expression: $(x - 2)^{1} = 4^{2}$.

Step 2.2.1.1.2.2:

Rewrite without the exponent: $x - 2 = 4^{2}$.

Step 2.2.1.2:

Conclude the simplification: $x - 2 = 4^{2}$.

Step 2.3:

Simplify the right-hand side.

Step 2.3.1:

Calculate $4^{2}$: $x - 2 = 16$.

Step 3:

Isolate $x$ by moving constants to the other side.

Step 3.1:

Add $2$ to both sides: $x = 16 + 2$.

Step 3.2:

Combine like terms: $x = 18$.

Knowledge Notes:

To solve an equation involving a square root, the following knowledge points are relevant:

1. Isolating the Radical: To solve for the variable within a radical, the first step is often to isolate the radical on one side of the equation.

2. Squaring Both Sides: Squaring both sides of an equation is a common technique to remove a square root. This must be done carefully to avoid introducing extraneous solutions.

3. Simplifying Expressions: Simplifying algebraic expressions involves combining like terms and reducing expressions to their simplest form.

4. Exponent Rules: Understanding how to manipulate exponents is crucial. For example, $(a^{m})^{n} = a^{mn}$ is the power of a power rule, which states that when raising an exponent to another power, you multiply the exponents.

5. Solving Linear Equations: After simplifying the equation, you often end up with a linear equation in the form $ax + b = c$, which can be solved by isolating $x$ through inverse operations (e.g., addition or subtraction).

6. Checking Solutions: After finding a potential solution, it's important to check it by substituting it back into the original equation to ensure it does not result in an undefined expression or a false statement.