Problem 64 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Rationalize the denominator of the expression. $$ \frac{9+\sqrt{2}}{3-\sqrt{2}} $$

Step-by-Step Solution

Answer

The rationalized form of the given expression is \(\frac{29 + 12\sqrt{2}}{11}\).

Step 1: Identify the conjugate of the denominator

The conjugate of the denominator can be found by changing the sign of the expression with the square root term. In our case, the denominator is \(3 - \sqrt{2}\), so the conjugate is \(3 + \sqrt{2}\).

Step 2: Multiply the numerator and the denominator by the conjugate

Now, we will multiply the numerator and the denominator of the given expression by the conjugate of the denominator. Remember to multiply both, so that we are effectively multiplying the whole expression by 1 and not changing its value. $$ \frac{9+\sqrt{2}}{3-\sqrt{2}} \cdot \frac{3+\sqrt{2}}{3+\sqrt{2}} $$

Step 3: Multiply the numerators and denominators

Next, we'll multiply the numerators with numerators and the denominators with denominators. Numerator: $$ (9+\sqrt{2})(3+\sqrt{2}) $$ Denominator: $$ (3-\sqrt{2})(3+\sqrt{2}) $$

Step 4: Simplify the numerator and the denominator

Let's simplify both the numerator and the denominator separately. Numerator: $$ (9+\sqrt{2})(3+\sqrt{2}) = 9\cdot3 + 9\cdot\sqrt{2} + \sqrt{2}\cdot3 + \sqrt{2}\cdot\sqrt{2} \\ = 27 + 9\sqrt{2} + 3\sqrt{2} + 2 = 29 + 12\sqrt{2} $$ Denominator: $$ (3-\sqrt{2})(3+\sqrt{2}) = 3\cdot3 + 3\cdot(-\sqrt{2}) + (-\sqrt{2})\cdot3 + (-\sqrt{2})(\sqrt{2}) \\ = 9 - 3\sqrt{2} - 3\sqrt{2} + 2 = 11 - 6\sqrt{2} + 6\sqrt{2} $$ Notice that the \( -6\sqrt{2} \) and \( 6\sqrt{2} \) in the denominator cancel each other out, leaving us with: $$ = 11 $$

Step 5: Write the simplified expression

Now that we have simplified both the numerator and the denominator, we can write the final expression. $$ \frac{29+12\sqrt{2}}{11} $$ So, the rationalized form of the given expression is \(\frac{29 + 12\sqrt{2}}{11}\).

Key Concepts

Conjugate in Algebra

In algebra, a conjugate refers to a pair of expressions that differ only by the sign between two terms. Suppose you have an expression of the form \( a - b \sqrt{c} \); its conjugate would be \( a + b \sqrt{c} \). The purpose of using the conjugate is to eliminate irrational numbers from the denominator of a fraction—usually simplifying calculations and resulting in cleaner mathematical expressions.

• Use in Rationalization: By multiplying the numerator and the denominator by the conjugate of the denominator, we make the denominator a rational number.
• Simplicity of Expression: Multiplying conjugates leads to differences of squares, which are simpler to handle.

Understanding how and when to use the conjugate can greatly facilitate solving expressions involving square roots.

Simplifying Square Roots

Square roots can sometimes make expressions look confusing or complex. Simplifying square roots helps us work with more manageable numbers. For example, simplifying \( \sqrt{8} \) gives us \( 2\sqrt{2} \). This is done by factoring the number under the square root into its prime factors and then "taking out" pairs of prime factors.

• Factoring: Break down the number under the root into its prime components.
• Pairing: For every pair of the same number, take one outside the root.

By simplifying square roots, you make the expression easier to understand and work with, especially when combining or comparing them with other similar expressions.

Algebraic Fractions

Algebraic fractions involve expressions where both the numerator and the denominator are polynomials or include variables with constants. These are similar to numerical fractions but involve a bit more complexity due to variables.

• Simplification: Algebraic fractions can often be simplified by factoring and canceling common factors in both the numerator and the denominator.
• Finding the Least Common Denominator (LCD): Sometimes, it is necessary to add or subtract algebraic fractions, and this requires finding the LCD.

The simplification and operation of algebraic fractions often bear similarities to regular fractions, but one must account for the additional complexities introduced by variables.

Mathematical Expressions

Mathematical expressions are combinations of numbers, operators, and variables. They are the building blocks of algebra and help represent complex ideas in a concise manner.

• Components: Can include numbers, letters (variables), and operations like addition, subtraction, multiplication, and division.
• Evaluating Expressions: To evaluate an expression means to calculate its value given the values of its variables.

Understanding and manipulating mathematical expressions is crucial for solving various algebraic problems. They allow for the abstraction of mathematical ideas, facilitating problem-solving and the representation of real-world situations.