Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Rationalize the denominator. $$\frac{3}{8+\sqrt{11}}$$

Step-by-Step Solution

Answer

Upon simplification, the resulting expression after rationalizing the denominator is \( \frac{3(8 - \sqrt{11})}{53}\).

Step 1: Identify the Conjugate

Firstly, identify the conjugate of the denominator. The conjugate of a number of the form \(a + \sqrt{b}\) is \(a - \sqrt{b}\), just changing the sign of the square root term. Identified conjugate is \(8 - \sqrt{11}\).

Step 2: Multiply with the Conjugate

Next, multiply both the numerator and the denominator by the conjugate.So, this fraction becomes \( \frac{3(8 - \sqrt{11})}{(8 + \sqrt{11}) (8 - \sqrt{11})}\).

Step 3: Simplify the Expression

Now, simplify the fraction. In the denominator we have the form \((a+b)(a-b)\), which simplifies to \(a^2 - b^2\).So, the fraction becomes \( \frac{3(8 - \sqrt{11})}{8^2 - (\sqrt{11})^2}\), which further simplifies to \( \frac{3(8 - \sqrt{11})}{64 - 11}\).

Key Concepts

Conjugate of a Number

When working with complex numbers or expressions with radicals, understanding the conjugate is key to simplification. The conjugate of a number mirrors a given number with the sign between its terms reversed.

For a number in the form of \(a + \(b\)\), its conjugate would be \(a - \(b\)\). The conjugate comes into play when we want to eliminate radicals from an equation or expression—especially from a denominator, because having a radical in the denominator is typically considered less simplified or standard.

Why is this useful? When you multiply an expression by its conjugate, the radical terms cancel out due to the difference of squares, a fundamental algebraic identity. This leaves you with a polynomial expression, which is generally easier to work with. For example, multiplying \((a + \(b\))(a - \(b\))\) results in \(a^2 - b^2\), which no longer contains the radical.

Simplifying Radical Expressions

The process of simplifying radical expressions is essential to make sure the expressions are in their most digestible form. Simplification typically involves removing the radicals, or at least reducing the index and radicand to its smallest possible numbers.

One of the most common techniques is to rationalize the denominator, which means to eliminate radicals from the bottom of a fraction. With expressions like \(\frac{3}{8+\sqrt{11}}\), you can achieve simplification by multiplying both the numerator and the denominator by the conjugate of the denominator. The goal is not just to make the expression 'look nicer', but to facilitate further operations, such as addition or multiplication with similar expressions. In our mathematical journey, we exploit identities like the difference of squares to transform these expressions into ones that are easier to manage, both for hand calculations and for further algebraic manipulation.

Multiplying Binomials

When it comes to multiplying two binomials, or two-term expressions, we are juggling with a multiplication strategy that may remind you of something you already know—the FOIL method, which stands for First, Outer, Inner, Last.

This technique is used to systematically multiply each term of one binomial with each term of the other. Consider the binomials \(a + b\) and \(c + d\). Their product is \(ac + ad + bc + bd\), the result of adding up the products of the First terms, Outer terms, Inner terms, and Last terms.

Getting comfortable with multiplying binomials is not just about rote learning, but about recognizing patterns that allow us to simplify expressions and solve equations that would otherwise be intimidating. This process establishes a foundation that prepares us for higher-level algebra, calculus, and other mathematical adventures.