Problem 55 - Algebra for College Students

Question

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{3}{\sqrt{2}-5}\)

Step-by-Step Solution

Answer

\(\frac{3\sqrt{2} + 15}{-23}\)

Step 1: Identify the Conjugate

To rationalize the denominator, we need to eliminate the square root in the denominator. We do this by multiplying by the conjugate, which is the same expression as the denominator but with the opposite sign. For the denominator \(\sqrt{2} - 5\), the conjugate is \(\sqrt{2} + 5\).

Step 2: Multiply Numerator and Denominator by Conjugate

Multiply both the numerator and the denominator by the conjugate \(\sqrt{2} + 5\). The expression becomes:\[\frac{3}{\sqrt{2} - 5} \times \frac{\sqrt{2} + 5}{\sqrt{2} + 5} = \frac{3(\sqrt{2} + 5)}{(\sqrt{2} - 5)(\sqrt{2} + 5)}\]

Step 3: Simplify the Denominator

The denominator is a difference of squares and simplifies as follows:\[(\sqrt{2} - 5)(\sqrt{2} + 5) = (\sqrt{2})^2 - 5^2 = 2 - 25 = -23\]

Step 4: Distribute in the Numerator

Expand the numerator by distributing \(3\) over \(\sqrt{2} + 5\):\[3(\sqrt{2} + 5) = 3\sqrt{2} + 15\]

Step 5: Combine and Simplify the Expression

Combine the expanded numerator with the simplified denominator:\[\frac{3\sqrt{2} + 15}{-23}\]This is the simplified expression with the denominator rationalized.

Key Concepts

Conjugate in Algebra

In algebra, a conjugate is a vital tool for rationalizing the denominator when faced with a radical. To understand this better, think about two terms containing a square root, such as

• the original term: \(\sqrt{a} + b\)
• its conjugate: \(\sqrt{a} - b\)

You'll notice that the only change is the sign between the terms. By multiplying the original expression by its conjugate, any radical part in the denominator can be effectively removed. This works well due to a pattern called the "difference of squares," which ultimately simplifies the expression.

Simplifying Expressions

Simplifying expressions means making them as straightforward as possible without changing their value. Using multiplication with conjugates helps remove radicals from denominators, simplifying these expressions significantly.
When you multiply the numerator and the denominator by the same conjugate, the value of the original expression remains unchanged, because this multiplication essentially represents multiplying by one:

• For example: \(\frac{3}{\sqrt{2} - 5} \times \frac{\sqrt{2} + 5}{\sqrt{2} + 5}\)
• This equals \(\frac{3(\sqrt{2} + 5)}{(\sqrt{2} - 5)(\sqrt{2} + 5)}\), which looks complicated at first but simplifies nicely.

The numerator expands to simpler terms by the distributive property, and the denominator simplifies using the difference of squares.

Difference of Squares

The difference of squares is a helpful algebraic pattern used when two squares are separated by a subtractive sign. It is expressed as:
\[(a + b)(a - b) = a^2 - b^2\]
This formula helps in rationalizing denominators that contain radicals. By identifying the difference of squares, the process of simplification becomes straightforward.
For instance, with our expression, the denominator uses:

• \((\sqrt{2} - 5)(\sqrt{2} + 5)\)
• This simplifies to \((\sqrt{2})^2 - 5^2 = 2 - 25 = -23\).

Using difference of squares lets us quickly rationalize and simplify expressions, ensuring all square roots are eliminated effectively.