Problem 367 - Elementary Algebra

Question

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{3}}{\sqrt{m}-\sqrt{5}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{3m} + \sqrt{15}}{m - 5} \)

Step 1: Understand the Problem

We need to simplify the expression \(\frac{\sqrt{3}}{\sqrt{m}-\sqrt{5}}\) by rationalizing the denominator.

Step 2: Identify the Conjugate of the Denominator

The denominator is \( \sqrt{m}-\sqrt{5} \). The conjugate of this expression is \( \sqrt{m}+\sqrt{5} \).

Step 3: Multiply Numerator and Denominator by the Conjugate

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

Step 4: Expand the Numerator

Using distribution, the numerator becomes:\[ \sqrt{3} \times (\sqrt{m} + \sqrt{5}) = \sqrt{3m} + \sqrt{15}.\]

Step 5: Simplify the Denominator

Using the difference of squares formula, \((a-b)(a+b) = a^2 - b^2\), the denominator becomes:\[ (\sqrt{m})^2 - (\sqrt{5})^2 = m - 5.\]

Step 6: Combine the Simplified Numerator and Denominator

The simplified form of the expression is:\[ \frac{\sqrt{3m} + \sqrt{15}}{m - 5}.\]

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra means making them easier to work with or understand. It often involves combining like terms, breaking down complex terms, and reducing fractions to their simplest form. For the given problem, we simplify the expression by rationalizing the denominator. This makes the denominator a rational number instead of an irrational number which is easier for further calculations. By simplifying:

• We make expressions cleaner and more manageable.
• We can often make additional calculations faster and more accurate.

The simplification process can be particularly useful in solving algebraic equations that involve radicals and fractions.

Conjugates in Algebra

In algebra, conjugates are pairs of binomials that are the same except for the sign between their terms. For example, \( \sqrt{a} + \sqrt{b} \) and \( \sqrt{a} - \sqrt{b} \) are conjugates of each other.

• When we multiply conjugates, we use the difference of squares formula which simplifies the expression.
• Conjugates are essential in rationalizing denominators with radicals.

In the step-by-step solution, we rationalized the denominator by multiplying both the numerator and denominator by the conjugate \( \sqrt{m} + \sqrt{5} \). This helps to eliminate the radicals in the denominator, making it easier to work with in further calculations.

Difference of Squares

The difference of squares is a powerful algebraic identity and states that \( (a - b)(a + b) = a^2 - b^2 \). This identity is especially useful when working with conjugates.

• In the context of rationalizing denominators, it helps in eliminating the radical terms in the denominator.
• This identity simplifies the multiplication process of conjugates.

In our example:

• First, identify the conjugates \( \sqrt{m} - \sqrt{5} \) and \( \sqrt{m} + \sqrt{5} \).
• Use the difference of squares formula: \( (\sqrt{m})^2 - (\sqrt{5})^2 = m - 5 \).

By leveraging the difference of squares, we simplified the denominator in the problem to \( m - 5 \), enabling a cleaner and more straightforward expression.