Problem 90 - A Graphical Approach to College Algebra

Question

Rationalize the denominator of each radical expression. Assume that all variables represent nonnegative real numbers and that no denominators are \(0 .\) $$\frac{3 m}{2+\sqrt{m+n}}$$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{6m - 3m\sqrt{m+n}}{4 - m - n} \).

Step 1: Identify the Conjugate

The expression given is \( \frac{3m}{2+\sqrt{m+n}} \). To rationalize the denominator, we need to eliminate the square root. We do this by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + \sqrt{m+n}\) is \(2 - \sqrt{m+n}\).

Step 2: Multiply by the Conjugate

Multiply the numerator and the denominator by the conjugate \(2 - \sqrt{m+n}\): \[\frac{3m}{2+\sqrt{m+n}} \times \frac{2-\sqrt{m+n}}{2-\sqrt{m+n}} = \frac{3m(2-\sqrt{m+n})}{(2+\sqrt{m+n})(2-\sqrt{m+n})}.\]

Step 3: Simplify the Denominator

The denominator \((2+\sqrt{m+n})(2-\sqrt{m+n})\) is a difference of squares. Use the identity \(a^2 - b^2 = (a+b)(a-b)\) with \(a = 2\) and \(b = \sqrt{m+n}\):\[(2)^2 - (\sqrt{m+n})^2 = 4 - (m+n) = 4 - m - n.\]

Step 4: Simplify the Numerator

Expand the numerator \(3m(2-\sqrt{m+n})\): \[3m \times 2 - 3m \times \sqrt{m+n} = 6m - 3m\sqrt{m+n}.\]

Step 5: Combine the Simplified Parts

Substitute the simplified numerator and denominator back into the expression:\[\frac{6m - 3m\sqrt{m+n}}{4 - m - n}.\]This is the expression with the denominator rationalized.

Key Concepts

Conjugate

In math, a conjugate is used to eliminate square roots or imaginary numbers from an expression. Specifically, it's a helpful tool in rationalizing denominators—making them easier to handle and understand. In our example, we have the denominator \(2 + \sqrt{m+n}\). By multiplying by its conjugate, \(2 - \sqrt{m+n}\), we effectively get rid of the square root.

Conjugates work based on a simple principle: \((a + b)(a - b) = a^2 - b^2\). Here, \(a = 2\) and \(b = \sqrt{m+n}\).

Key takeaways:

• The conjugate of \(a + b\) is \(a - b\), and vice versa.
• It helps turn irrational expressions into rational ones.
• By applying the conjugate, complex fractions become simpler to work with.

Understanding conjugates can simplify complex algebraic expressions significantly by eliminating inconvenient radicals or imaginary parts.

Difference of Squares

The difference of squares is a widely used algebraic identity, important for simplifying expressions when dealing with conjugates. This identity follows the formula: \[a^2 - b^2 = (a+b)(a-b)\].
This formula is essential when you have expressions like \((2 + \sqrt{m+n})(2 - \sqrt{m+n})\).

By applying this identity, the expression transforms into something simpler:

• Calculate \((2)^2 - (\sqrt{m+n})^2\).
• This becomes \(4 - (m+n)\).
• The result is easily computed: \(4 - m - n\).

Difference of squares formula effectively cancels out the radical terms in the denominator, leaving us with just numbers or simpler expressions.
This is why it's a powerful technique for rationalizing denominators.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that makes working with equations far more manageable. It involves reducing the expression to its simplest form.

In our solution:

• We simplify the numerator by multiplying: \(3m(2 - \sqrt{m+n})\).
• Distribute \(3m\) across each term: \(6m - 3m\sqrt{m+n}\).
• With a simplified numerator and denominator, the expression becomes clearer: \(\frac{6m - 3m\sqrt{m+n}}{4 - m - n}\).

Simplifying helps ensure that the expression is as clear and concise as possible.
This makes further operations, like solving or substituting, less prone to error and more intuitive. Keeping expressions simple is not just an arbitrary rule; it enhances understanding and efficiency in solving problems.