Simplify (x-2)/( square root of x^2-4)
The problem requires you to simplify the given algebraic expression. The expression consists of a numerator (x-2) and a denominator which is the square root of a binomial expression (x^2 - 4). The goal is to manipulate the expression in such a way that it becomes as straightforward as possible, possibly by factoring, canceling terms, or using algebraic identities, so that no further simplification is possible.
$\frac{x - 2}{\sqrt{x^{2} - 4}}$
Answer
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:
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.
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.
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.
Square Roots: The square root of a product is equal to the product of the square roots of each factor, that is, $\sqrt{ab} = \sqrt{a}\sqrt{b}$.
Simplifying Expressions: Simplifying an expression involves combining like terms, canceling common factors, and applying arithmetic operations to reach the simplest form of the expression.
