Problem 62 - Algebra for College Students

Question

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{\sqrt{7}}{3 \sqrt{2}-5}\)

Step-by-Step Solution

Answer

The expression rationalized and simplified is \(-\frac{3 \sqrt{14} + 5 \sqrt{7}}{7}\).

Step 1: Identify the Conjugate

The given expression is \( \frac{\sqrt{7}}{3 \sqrt{2} - 5} \). To rationalize the denominator, identify the conjugate of \( 3 \sqrt{2} - 5 \), which is \( 3 \sqrt{2} + 5 \).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \( 3 \sqrt{2} + 5 \): \[\frac{\sqrt{7}}{3 \sqrt{2} - 5} \times \frac{3 \sqrt{2} + 5}{3 \sqrt{2} + 5} = \frac{\sqrt{7}(3 \sqrt{2} + 5)}{(3 \sqrt{2} - 5)(3 \sqrt{2} + 5)}\]

Step 3: Expand the Numerator

Expand the expression in the numerator: \[\sqrt{7}(3 \sqrt{2} + 5) = 3 \sqrt{14} + 5 \sqrt{7}\].

Step 4: Use the Difference of Squares for the Denominator

The denominator uses the identity \((a - b)(a + b) = a^2 - b^2\) where \(a = 3 \sqrt{2} \) and \(b = 5\):\[\begin{aligned}(3 \sqrt{2})^2 - 5^2 &= 9 \cdot 2 - 25 \&= 18 - 25 \&= -7.\end{aligned}\]

Step 5: Write the Final Expression

Now substitute back into the fraction: \[\frac{3 \sqrt{14} + 5 \sqrt{7}}{-7} = -\frac{3 \sqrt{14} + 5 \sqrt{7}}{7}\].

Key Concepts

Conjugates

When dealing with expressions that include square roots, particularly in the denominator, conjugates are very useful. A conjugate is formed by changing the sign between two terms. For example, if the original expression is \( a - b \), its conjugate would be \( a + b \). In our exercise, we started with \( 3 \sqrt{2} - 5 \) and found its conjugate to be \( 3 \sqrt{2} + 5 \).

Using conjugates helps because when you multiply a term by its conjugate, it eliminates the radical in the denominator. This simplification is possible due to the difference of squares identity, which we will explore next. This process makes the expression easier to work with, reduces complexity, and is thus encouraged in mathematical rationalizations.

Difference of Squares

The difference of squares is a powerful algebraic identity used to simplify expressions, especially when rationalizing denominators. The identity can be expressed as:

• \((a + b)(a - b) = a^2 - b^2\)

In the context of our exercise, by multiplying the original denominator \(3 \sqrt{2} - 5\) with its conjugate \(3 \sqrt{2} + 5\), we apply this rule:

• \((3 \sqrt{2})^2 - 5^2 \)
• \(= 9 \times 2 - 25\)
• \(= 18 - 25\)
• \(= -7\)

This calculation effectively removes the square root from the denominator, resulting in a simplified rational expression.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra that helps in achieving more straightforward, manageable forms of mathematical expressions. In the process of rationalizing the denominator, simplification occurs in several steps.

After identifying and multiplying by the conjugate, the next step is expanding and combining like terms in the numerator. For the numerator expression \(\sqrt{7} \times (3 \sqrt{2} + 5)\), we distribute the \(\sqrt{7}\) to get:

• \(3 \sqrt{14} + 5 \sqrt{7}\)

This expansion step is crucial to represent the expression in a simpler form that can be used in further operations.

Subsequent simplifications are applied as necessary, including canceling terms and reducing fractions, if applicable, to achieve the final, simplest form.

Radicals and Square Roots

Radicals and square roots appear frequently in algebra, representing the root of a number. Working with these symbols requires understanding of a few essential operations. For instance, \(\sqrt{a}^2 = a\), which shows that the square root and the square are inverse operations.

In our exercise, we dealt with a radical expression in the denominator \(3 \sqrt{2} - 5\). The goal was to rationalize this by removing the square root, achieved by using its conjugate. Radicals can complicate expressions and calculations, but understanding the properties of square roots and strategies like using conjugates help simplify these problems effectively.

Always remember that manipulating radicals requires careful handling to maintain mathematical accuracy and integrity.