Problem 36 - Elementary Algebra

Question

For the following problems, simplify the expressions. $$ \frac{10}{a-\sqrt{2 b}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\frac{10}{a-\sqrt{2 b}}$$. Answer: The simplified expression is $$\frac{10(a+\sqrt{2b})}{a^2 - 2b}$$

Step 1: Identify the conjugate of the denominator

The conjugate of the denominator is \((a+\sqrt{2b})\). We will multiply the both the numerator and denominator by this factor.

Step 2: Multiply the numerator and denominator by the conjugate

Multiply the numerator and the denominator by the conjugate of the denominator \((a+\sqrt{2b})\): $$ \frac{10}{a-\sqrt{2 b}} \cdot \frac{a+\sqrt{2b}}{a+\sqrt{2b}} $$

Step 3: Multiply numerator and simplify

Multiply the factors in the numerator: $$ \frac{10(a+\sqrt{2b})}{(a-\sqrt{2 b})(a+\sqrt{2b})} $$

Step 4: Multiply denominator and simplify

Apply the difference of squares formula to the denominator: \((a-\sqrt{2 b})(a+\sqrt{2b}) = a^2 - (\sqrt{2b})^2\) $$ \frac{10(a+\sqrt{2b})}{a^2 - 2b} $$

Step 5: Final expression

The simplified expression is: $$ \frac{10(a+\sqrt{2b})}{a^2 - 2b} $$

Key Concepts

Conjugate of a Denominator

The concept of the conjugate of a denominator is an essential tool in simplifying algebraic fractions, particularly those involving square roots. A conjugate in algebra refers to a binomial formed by changing the sign between two terms. For instance, if we have a denominator like \(a - \sqrt{2b}\), its conjugate would be \(a + \sqrt{2b}\). When we multiply the original expression by a fraction with the conjugate as both its numerator and denominator, we don't change the value of the expression, because, in essence, we're multiplying by one. But what we achieve is a remarkable simplification.

This process removes the square root from the denominator, which is often a desirable outcome for the final answer to be in the standard form. This is because the product of a binomial and its conjugate leads to a difference of squares, which is an easier expression to deal with, neatly canceling out the square root terms.

Difference of Squares

The difference of squares is an algebraic pattern that's incredibly handy in simplifying expressions. When two squared terms are subtracted, such as \(a^2 - b^2\), the result factors neatly into \((a+b)(a-b)\). In our textbook exercise, when we multiply the denominator by its conjugate, we apply this pattern.

The denominator \((a-\sqrt{2 b})(a+\sqrt{2b})\) transforms into \(a^2 - (\sqrt{2b})^2\). Notice how the radical part is squared, which eliminates the square root, leading to the even simpler expression \(a^2 - 2b\). The difference of squares is a powerful tool for simplifying expressions because it often leads to clean, rational denominators and can reduce complex expressions to more manageable ones.

Rational Expressions

When we talk about rational expressions, we're referring to fractions where the numerator and denominator are both polynomials. The expression \(\frac{10}{a-\sqrt{2b}}\) is a rational expression with an irrational element, namely the square root. Simplifying such expressions often involves making the denominator rational whenever possible, so that the expression can be worked with more easily, including further operations like addition, subtraction, or comparison.

By using the conjugate of the denominator and the difference of squares technique—as shown in the exercise and solution—we create a rational expression where both the numerator and the denominator are free from radicals. This simplifies the process of performing any arithmetic operations and comparing rational expressions, facilitating understanding and manipulation of these algebraic fractions.