Problem 53 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{s+3}}{\sqrt{s-3}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\frac{\sqrt{s+3}}{\sqrt{s-3}}$$ Answer: After simplification, the expression becomes $$\frac{s+3}{6}$$.

Step 1: Identify the conjugate of the denominator

The given expression is $$\frac{\sqrt{s+3}}{\sqrt{s-3}}$$. The conjugate of the denominator, that is of $$\sqrt{s-3}$$, is $$\sqrt{s+3}$$. We will multiply both the numerator and the denominator by the conjugate to get rid of the square root in the denominator.

Step 2: Multiply by the conjugate

Now we can multiply the numerator and the denominator of \(\frac{\sqrt{s+3}}{\sqrt{s-3}}\) by \(\frac{\sqrt{s+3}}{\sqrt{s+3}}\): $$ \frac{\sqrt{s+3}}{\sqrt{s-3}} \cdot \frac{\sqrt{s+3}}{\sqrt{s+3}} = \frac{(\sqrt{s+3})^2}{(\sqrt{s-3})(\sqrt{s+3})} $$

Step 3: Simplify the numerator and denominator

Simplify the numerator and denominator: $$ \frac{(\sqrt{s+3})^2}{(\sqrt{s-3})(\sqrt{s+3})} = \frac{s+3}{(\sqrt{s-3})(\sqrt{s+3})} $$ Now, we apply the difference of squares formula for the denominator. We know that \(a^2 - b^2 = (a+b)(a-b)\). Here, \(a=\sqrt{s+3}\) and \(b=\sqrt{s-3}\). The denominator can be written as: $$ (\sqrt{s-3})(\sqrt{s+3}) = (\sqrt{s+3})^2 - (\sqrt{s-3})^2 $$

Step 4: Substitute the expression of the denominator

Now, substitute the expression of the denominator into the main expression: $$ \frac{s+3}{(\sqrt{s-3})(\sqrt{s+3})} = \frac{s+3}{(\sqrt{s+3})^2 - (\sqrt{s-3})^2} $$ Simplify further: $$ \frac{s+3}{s+3 - (s-3)} = \frac{s+3}{6} $$

Step 5: Write the final simplified expression

Now that we have simplified the expression, the final result is: $$ \frac{\sqrt{s+3}}{\sqrt{s-3}} = \frac{s+3}{6} $$

Key Concepts

Conjugate of a Square Root

When simplifying expressions involving square roots, you may encounter a situation where the denominator includes a square root, leaving the expression in an 'unsimplified' state. To rectify this, we use the method of multiplying by the conjugate of the square root in both the numerator and the denominator.

The conjugate of a square root, \textbf{for example, \( \sqrt{a} \)} is \( \sqrt{a} \) with the opposite sign, so \( \sqrt{a} \) has the conjugate \( -\sqrt{a} \), and vice versa.

By multiplying by the conjugate, you effectively eliminate the square root from the denominator. This technique is commonly used because it leverages the difference of squares pattern, which we will discuss in the next section.

Difference of Squares

Understanding the difference of squares formula is critical when working with radical expressions, especially when simplifying them. The formula \( a^2 - b^2 = (a+b)(a-b) \) is a fundamental identity in algebra that can be applied in various scenarios.

It is particularly helpful after multiplying an expression by the conjugate of a square root, because when you have terms like \( (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) \) it simplifies to \( a - b \) due to the nature of the difference of squares. This property is leveraged in the original example exercise to simplify the expression \( \frac{\sqrt{s+3}}{\sqrt{s-3}} \) by eliminating the radicals from the denominator.

Simplifying Square Roots

Simplifying square roots is a process of finding an equivalent expression where the square root is expressed in its simplest form. The method involves identifying and factoring out perfect squares from the radicand—the number under the square root sign—and simplifying the expression accordingly.

A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is \( 6^2 \) and can be simplified from \( \sqrt{36} \) to 6. When simplifying, you should look for such factors inside the square root to simplify the expression, much like what is done in the provided step by step solution, but with the additional step of multiplying by the conjugate.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. When working with these expressions it's essential to understand the order of operations and how to manipulate the terms to simplify or solve equations.

For simplification, you may combine like terms, factor expressions, and apply algebraic identities such as the distributive property or the difference of squares. In the context of simplifying square roots, the algebraic manipulations will often involve methods like factoring, expanding, and rationalizing denominators to reach a simplified form. These tasks require a careful approach to ensure that each step follows logically from the previous and adheres to the rules of algebra.