Problem 60 - Algebra for College Students

Question

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{3}}{\sqrt{7}-\sqrt{2}}\)

Step-by-Step Solution

Answer

\(\frac{\sqrt{21} + \sqrt{6}}{5}\).

Step 1: Identify the Conjugate

The first step is to identify the conjugate of the denominator \(\sqrt{7} - \sqrt{2}\). The conjugate is the expression \(\sqrt{7} + \sqrt{2}\). This is because the conjugate of \(a - b\) is \(a + b\).

Step 2: Multiply by the Conjugate

Next, multiply both the numerator and the denominator by the conjugate \(\sqrt{7} + \sqrt{2}\). This yields: \[ \frac{\sqrt{3}}{\sqrt{7} - \sqrt{2}} \times \frac{\sqrt{7} + \sqrt{2}}{\sqrt{7} + \sqrt{2}} = \frac{\sqrt{3}(\sqrt{7} + \sqrt{2})}{(\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2})} \]

Step 3: Apply the Difference of Squares Formula

Apply the difference of squares formula to simplify the denominator. This formula is \((a-b)(a+b) = a^2 - b^2\). So, \((\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5\).

Step 4: Simplify the Denominator

Using the result from Step 3, we have the denominator as \(5\). Therefore, our expression has simplified to: \[ \frac{\sqrt{3}(\sqrt{7} + \sqrt{2})}{5} \]

Step 5: Distribute the Numerator

Distribute the \(\sqrt{3}\) in the numerator: \(\sqrt{3}(\sqrt{7}) + \sqrt{3}(\sqrt{2}) = \sqrt{21} + \sqrt{6}\).

Step 6: Final Expression

The expression after simplifying both the numerator and denominator is: \[ \frac{\sqrt{21} + \sqrt{6}}{5} \] This is the fully simplified expression with a rational denominator.

Key Concepts

Conjugate in Algebra

Conjugates in algebra are an essential tool for simplifying expressions, especially when dealing with complex or irrational numbers. The concept revolves around the idea of pairs of binomial expressions that involve the sum and difference of the same values. In most cases, these values are square roots or radicals.

Here's how it works:

• If you have a binomial expression like \(a - b\), its conjugate will be \(a + b\).
• This means you're simply changing the sign between the two terms.

This technique is invaluable when rationalizing denominators, as it helps to eliminate radicals by utilizing the difference of squares formula. It's like untying a knot in an equation by balancing both sides without really changing the value of the expression.
The conjugate method isn't just useful for rationalizing expressions but is also applicable in various algebraic scenarios like solving equations and even in complex number operations.

Difference of Squares

The difference of squares is a fundamental algebraic identity that comes in handy when working with conjugates and rationalizing expressions. It states that the product of a sum and a difference of the same two terms can be expressed as the difference between their squares:

\[(a-b)(a+b) = a^2 - b^2\]
This is particularly useful because it allows us to remove radicals from the denominator, simplifying the expression.

• For example, if you encounter the expression \((\sqrt{7} - \sqrt{2})(\sqrt{7} + \sqrt{2})\), applying the difference of squares identity results in:
  \((\sqrt{7})^2 - (\sqrt{2})^2 = 7 - 2 = 5\).

By transforming a product of binomials into a simple subtraction of squares, expressions become far easier to work with, both numerically and algebraically.
This identity helps maintain order and structure, especially when simplifying expressions in calculus, trigonometry, and other mathematical disciplines.

Simplification of Radical Expressions

Simplifying radical expressions is a key process in algebra that often involves rationalizing the denominator to make calculations easier and expressions neater. The goal is to eliminate any irrational numbers from the denominator of a fraction.

Here's the procedure to follow:

• Find the conjugate of the denominator, then multiply both the numerator and denominator by this conjugate.
• Use the difference of squares formula to simplify the denominator.
• Expand the expression in the numerator if needed.

For instance, with \(\frac{\sqrt{3}}{\sqrt{7} - \sqrt{2}}\), multiplying by the conjugate \(\sqrt{7} + \sqrt{2}\) modifies the problem to:
\[\frac{\sqrt{3}(\sqrt{7} + \sqrt{2})}{5}\]
After expanding and simplifying the numerator, you end up with:
\[\frac{\sqrt{21} + \sqrt{6}}{5}\]
This expression is now rationalized with a neat denominator, ideal for further operations or calculations involving the expression. Remember, the simplification of radical expressions improves fluency in problem-solving, making algebraic manipulations more manageable and less prone to error.