Solution: Step 1: Express the number 9 as the square of 3. Thus, the expression becomes frac7x2 - 32. Step 2: Recognize that the denominator is a difference of squares. Apply the difference of squares identity, which is a2 - b2 = (a + b)(a - b), where in this case, a = x and b = 3. This simplifies the expression to frac7(x + 3)(x - 3). Knowledge Notes: The problem involves simplifying a rational expression with a denominator that is a difference of squares. The relevant knowledge points for solving this problem include: 1. Difference of Squares: This is a pattern that is recognized as a2 - b2, which can be factored into (a + b)(a - b). It is a basic algebraic identity that is often used to simplify expressions and solve equations. 2. Perfect Squares: Numbers like 9, which can be expressed as the square of an integer (in this case, 32), are called perfect squares. Recognizing perfect squares is useful when factoring expressions, especially when applying the difference of squares formula. 3. Factoring: This is the process of breaking down an expression into a product of simpler expressions. In this problem, factoring the denominator using the difference of squares formula simplifies the rational expression. 4. Simplifying Rational Expressions: The process of simplifying rational expressions often involves factoring the numerator and the denominator and then canceling out common factors. In this case, there are no common factors to cancel, but factoring the denominator is a necessary step in simplifying the expression. Understanding these concepts is crucial for simplifying algebraic expressions and solving a wide range of mathematical problems.

Simplify 7/(x^2-9)

The problem asks to perform a mathematical simplification on the algebraic expression "7/(x^2-9)". This involves reducing the expression to its simplest form, typically by factoring denominators and possibly canceling out common factors in the numerator and denominator if applicable. Simplification of algebraic expressions is done to make them easier to understand or to prepare them for further mathematical operations or solving.

$\frac{7}{x^{2} - 9}$

Answer

Expert–verified

Solution:

Step 1:

Express the number $9$ as the square of $3$ . Thus, the expression becomes $\frac{7}{x^{2} - 3^{2}}$ .

Step 2:

Recognize that the denominator is a difference of squares. Apply the difference of squares identity, which is $a^{2} - b^{2} = (a + b) (a - b)$ , where in this case, $a = x$ and $b = 3$ . This simplifies the expression to $\frac{7}{(x + 3) (x - 3)}$ .

Knowledge Notes:

The problem involves simplifying a rational expression with a denominator that is a difference of squares. The relevant knowledge points for solving this problem include:

Difference of Squares: This is a pattern that is recognized as $a^{2} - b^{2}$ , which can be factored into $(a + b) (a - b)$ . It is a basic algebraic identity that is often used to simplify expressions and solve equations.
Perfect Squares: Numbers like $9$ , which can be expressed as the square of an integer (in this case, $3^{2}$ ), are called perfect squares. Recognizing perfect squares is useful when factoring expressions, especially when applying the difference of squares formula.
Factoring: This is the process of breaking down an expression into a product of simpler expressions. In this problem, factoring the denominator using the difference of squares formula simplifies the rational expression.
Simplifying Rational Expressions: The process of simplifying rational expressions often involves factoring the numerator and the denominator and then canceling out common factors. In this case, there are no common factors to cancel, but factoring the denominator is a necessary step in simplifying the expression.

Understanding these concepts is crucial for simplifying algebraic expressions and solving a wide range of mathematical problems.