Simplify (x-2)/( square root of x^2-4)

Solution:

Step 1: Simplify the expression's denominator.

Step 1.1

Express $4$ as $2^2$. $$\frac{x-2}{\sqrt{x^2-4}}=\frac{x-2}{\sqrt{x^2-2^2}}$$

Step 1.2

Apply the difference of squares identity $a^2-b^2=(a+b)(a-b)$, where $a=x$ and $b=2$. $$\frac{x-2}{\sqrt{x^2-4}}=\frac{x-2}{\sqrt{(x+2)(x-2)}}$$

Step 2: Rationalize the denominator.

Multiply the expression by $\frac{\sqrt{(x+2)(x-2)}}{\sqrt{(x+2)(x-2)}}$. $$\frac{x-2}{\sqrt{(x+2)(x-2)}}\cdot \frac{\sqrt{(x+2)(x-2)}}{\sqrt{(x+2)(x-2)}}$$

Step 3: Combine and simplify the denominator.

Step 3.1

Multiply the numerators and denominators. $$\frac{(x-2)\sqrt{(x+2)(x-2)}}{\sqrt{(x+2)(x-2)}\sqrt{(x+2)(x-2)}}$$

Step 3.2

Raise the square root to the power of 1. $$\frac{(x-2)\sqrt{(x+2)(x-2)}}{(\sqrt{(x+2)(x-2)}{)}^1\sqrt{(x+2)(x-2)}}$$

Step 3.3

Combine the square roots in the denominator using the power rule $a^m{a}^n=a^{m+n}$. $$\frac{(x-2)\sqrt{(x+2)(x-2)}}{(\sqrt{(x+2)(x-2)}{)}^{1+1}}$$

Step 3.4

Add the exponents 1 and 1. $$\frac{(x-2)\sqrt{(x+2)(x-2)}}{(\sqrt{(x+2)(x-2)}{)}^2}$$

Step 3.5

Rewrite the square of the square root as the original expression. $$\frac{(x-2)\sqrt{(x+2)(x-2)}}{(x+2)(x-2)}$$

Step 4: Cancel the common factor of $x-2$.

Step 4.1

Cancel out the common factor. $$\frac{\cancel{(x-2)}\sqrt{(x+2)(x-2)}}{(x+2)\cancel{(x-2)}}$$

Step 4.2

The final simplified expression is: $$\frac{\sqrt{(x+2)(x-2)}}{x+2}$$

Knowledge Notes:

1. Difference of Squares: The difference of squares formula is $a^2-b^2=(a+b)(a-b)$. It is used to factor expressions where two terms are perfect squares separated by a subtraction sign.

2. Rationalizing the Denominator: To rationalize a denominator means to eliminate any radical expressions (like square roots) from the denominator. This is often done by multiplying the numerator and denominator by a suitable form of 1, such as the radical itself.

3. Power Rule: The power rule for exponents states that $a^m{a}^n=a^{m+n}$. This is used to combine like bases with different exponents.

4. Square Roots: The square root of a product is equal to the product of the square roots of each factor, that is, $\sqrt{ab}=\sqrt{a}\sqrt{b}$.

5. Simplifying Expressions: Simplifying an expression involves combining like terms, canceling common factors, and applying arithmetic operations to reach the simplest form of the expression.