Problem 81 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \frac{8-\sqrt{3}}{2+\sqrt{18}} $$

Step-by-Step Solution

Answer

#Question# Simplify the given expression: \(\frac{8-\sqrt{3}}{2+\sqrt{18}}\) #Answer# \(\frac{16 - 8\sqrt{18} - 2\sqrt{3} + 3\sqrt{6}}{-14}\)

Step 1: Identifying the conjugate of the denominator

The given expression is \(\frac{8-\sqrt{3}}{2+\sqrt{18}}\). We need to rationalize the denominator to eliminate the square root symbol. To do so, we can multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of an expression is simply the same expression with switched signs when involving two terms. So, the conjugate of the denominator, \(2+\sqrt{18}\), would be \(2-\sqrt{18}\).

Step 2: Multiply the numerator and denominator by the conjugate

Now, we can multiply both the numerator and the denominator by the conjugate of the denominator, \(2-\sqrt{18}\). It will practically look like multiplying by 1, keeping the value of the expression the same: $$ \frac{8-\sqrt{3}}{2+\sqrt{18}} \times \frac{2-\sqrt{18}}{2-\sqrt{18}} $$

Step 3: Perform the multiplication for numerator and denominator

We need to now perform the multiplication of both the numerator and the denominator. Use the distributive property to multiply each term of the numerator with the entire conjugate, and then use the difference of squares property for the denominator. Here's the calculation: Numerator: $$ (8-\sqrt{3}) (2-\sqrt{18}) = (8\times 2) - (8\times \sqrt{18}) - (\sqrt{3}\times 2) + (\sqrt{3}\times \sqrt{18}) $$ Denominator: $$ (2+\sqrt{18})(2-\sqrt{18}) = 2^2 - (\sqrt{18})^2 = 4 - 18 $$

Step 4: Simplify the expression

Now, let's simplify the numerator and denominator. The numerator simplifies to: $$ 16 - 8\sqrt{18} - 2\sqrt{3} + 3\sqrt{6} $$ The denominator simplifies to: $$ -14 $$ So, the simplified expression is: $$ \frac{16 - 8\sqrt{18} - 2\sqrt{3} + 3\sqrt{6}}{-14} $$

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a technique used in algebra to eliminate irrational numbers (like square roots) from the bottom part of a fraction. This is done to achieve a more simplified and standard form, often for the sake of easier calculation or clarity. For example, if you encounter a fraction such as \( \frac{8-\sqrt{3}}{2+\sqrt{18}} \), you may find it challenging to further simplify or evaluate due to the square root in the denominator.

How do we resolve this? We multiply the fraction by a clever form of 1: the conjugate of the denominator over itself. This doesn't change the value of the expression but conveniently alters its form. When we multiply \( 2+\sqrt{18} \) with its conjugate \( 2-\sqrt{18} \), we utilize a key property of conjugates in algebra—the difference of squares—a method that simplifies the expression and effectively removes the radical from the denominator.

Conjugate in Algebra

In algebra, the conjugate refers to a binomial that is exactly the same as another binomial, except that the sign between the two terms is opposite. For example, the conjugate of \( a+b \) is \( a-b \), and vice versa. It's particularly useful when dealing with complex expressions, especially when you have a binomial with a radical (like a square root) and a rational number.

Importance of Using the Conjugate

The use of conjugates is essential when rationalizing the denominator, as they help in removing radicals from the denominator. When you multiply a binomial by its conjugate, you will always end up with a difference of squares, which significantly simplifies the expression. This procedure is especially valuable in solving equations, simplifying expressions, and even in calculus.

Difference of Squares

The difference of squares is a fundamental algebraic pattern that emerges when you subtract one perfect square from another. In mathematical terms, it is expressed as \( a^2 - b^2 \), which can be factored into \( (a+b)(a-b) \). This is essential in many areas of algebra, including the multiplication and division of binomials and the aforementioned process of rationalizing the denominator.

When dealing with rationalizing the denominator, once the conjugate is multiplied, the result in the denominator is always the difference of squares, thus eliminating the radical and simplifying the expression. For the exercise mentioned, \( (2+\sqrt{18})(2-\sqrt{18}) \) simplifies to \( 4 - 18 \), a clear use of the difference of squares. Recognizing and applying the difference of squares simplifies complex algebraic expressions and is a key skill in advancing through algebra.