Problem 71 - Elementary Algebra

Question

Simplify each expression by performing the indicated operation. $$ \frac{4}{6+\sqrt{2}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $$\frac{4}{6+\sqrt{2}}$$. Answer: $$\frac{24-4\sqrt{2}}{34}$$

Step 1: Rationalize the denominator

To rationalize the denominator, we multiply both numerator and denominator by the conjugate of 6 + √2, which is 6 - √2. So, we have: $$ \frac{4}{6+\sqrt{2}} \times \frac{6-\sqrt{2}}{6-\sqrt{2}} $$

Step 2: Multiply numerators and denominators

Now, we multiply the numerators together and the denominators together: $$ \frac{4 \times (6-\sqrt{2})}{(6+\sqrt{2})(6-\sqrt{2})} $$

Step 3: Simplify the numerator

Simplify the numerator by distributing 4 to both terms: $$ \frac{24-4\sqrt{2}}{(6+\sqrt{2})(6-\sqrt{2})} $$

Step 4: Simplify the denominator

Simplify the denominator using the difference of squares formula \((a+b)(a-b) = a^2 - b^2\). In this case, \(a=6\) and \(b=\sqrt{2}\): $$ \frac{24-4\sqrt{2}}{6^2 - (\sqrt{2})^2} $$ Now substitute the values and simplify further: $$ \frac{24-4\sqrt{2}}{36 - 2} $$ $$ \frac{24-4\sqrt{2}}{34} $$ Now the expression is simplified, and the final result is: $$ \frac{24-4\sqrt{2}}{34} $$

Key Concepts

Difference of Squares

The difference of squares is a powerful tool in mathematics, often used to simplify expressions. It is a recognizable pattern that emerges when we subtract two perfect squares. In algebra, it is expressed as \( (a+b)(a-b) = a^2 - b^2 \).This pattern is useful for simplifying products of binomials since it allows us to quickly calculate the difference of two square numbers without expanding the entire expression.

Here's how it works:

• Identify two terms, each a square: in our exercise, we have \( a = 6 \) and \( b = \sqrt{2} \).
• Using the difference of squares, the expression \( (6+\sqrt{2})(6-\sqrt{2}) \) simplifies to \( 6^2 - (\sqrt{2})^2 \).

Our result was \( 36 - 2 \), simplifying to \( 34 \). This result makes computations more direct and less cluttered, which is highly effective in algebraic manipulation.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This step is crucial for making mathematical expressions easier to understand and solve. It involves combining like terms, reducing fractions, and eliminating radicals when possible.

In our example:

• We multiplied to remove the irrational part from the denominator.
• We simplified both the numerator and denominator using algebraic techniques.

By doing these actions, the expression \( \frac{4}{6+\sqrt{2}} \) became \( \frac{24-4\sqrt{2}}{34} \), making it more manageable. This process helps in isolating variables, solving equations, or preparing the expression for further operations.

Radical Expressions

Radical expressions involve roots, often providing challenging steps in simplification. Understanding radicals, especially square roots, is key to handling these expressions effectively. Simplification consists of rewriting the expression without a radical in the denominator, known as rationalizing.

In the original exercise, the term \( \sqrt{2} \) appeared in the denominator. Since it's not considered simplified when radicals are present in the denominator, rationalizing by multiplying by its conjugate was necessary.

• The conjugate used was \( 6 - \sqrt{2} \) to effectively eliminate the radical.
• Multiplying both numerator and denominator by this conjugate resulted in a simplified rational expression.

Ultimately, mastering radical expressions simplifies complicated numbers and aids in understanding more complex algebraic expressions, providing a clearer path to the solution.