Problem 64 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The rationalized and simplified form is \(-\frac{10\sqrt{5}}{43} + \frac{15\sqrt{7}}{43}\).

Step 1: Identify the Conjugate

To rationalize the denominator, we use the conjugate of the denominator. Here, the denominator is \(2\sqrt{5} + 3\sqrt{7}\). The conjugate of this expression is \(2\sqrt{5} - 3\sqrt{7}\).

Step 2: Multiply by the Conjugate

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

Step 3: Apply the Difference of Squares

The denominator is now a difference of squares: \((2\sqrt{5})^2 - (3\sqrt{7})^2\). Calculating each, we get: \[(2\sqrt{5})^2 = 4 \times 5 = 20\] \[(3\sqrt{7})^2 = 9 \times 7 = 63\]Thus, the difference is: \[20 - 63 = -43\]

Step 4: Simplify the Numerator

Distribute the 5 in the numerator: \[5 \times (2\sqrt{5} - 3\sqrt{7}) = 10\sqrt{5} - 15\sqrt{7}\]

Step 5: Form the Rationalized Expression

Combine the simplified numerator and denominator:\[\frac{10\sqrt{5} - 15\sqrt{7}}{-43}\]

Step 6: Simplify the Fraction

Since there are no common factors between the terms, the expression is fully simplified as:\[-\frac{10\sqrt{5}}{43} + \frac{15\sqrt{7}}{43}\]

Key Concepts

Conjugate in Algebra

When dealing with expressions involving square roots, especially in the denominator, we can simplify them using a concept called the *conjugate*. The conjugate of an expression like \(a + b\) is \(a - b\), and vice versa. In algebra, these are used to remove radicals (i.e., square roots) from the denominator, making the expression easier to manage and compute.

• Purpose: The conjugate helps in eliminating radicals from the denominator when dividing expressions.
• How to Use: To rationalize the denominator of \( \frac{c}{a + b} \), multiply both the numerator and the denominator by the conjugate \(a - b\).
• Outcome: This process transforms the original denominator into a more simplified form without radicals.

Recognizing and utilizing conjugates effectively allows you to manipulate and simplify algebraic expressions, which is particularly important in preparing for more advanced calculations.

Difference of Squares

The difference of squares is a powerful algebraic tool often employed in conjunction with conjugates. It follows the identity: \[ (a + b)(a - b) = a^2 - b^2 \]This identity is called the *difference of squares* because it results in the "difference" between two squared terms.

• Application: We use it to simplify products of conjugates when rationalizing denominators.
• Example: For the expression \( (2\sqrt{5} + 3\sqrt{7})(2\sqrt{5} - 3\sqrt{7}) \), using the identity gives us:
  • \((2\sqrt{5})^2 = 20\)
  • \((3\sqrt{7})^2 = 63\)
  • The difference: \(20 - 63 = -43\)

The difference of squares is fundamental when working with conjugates as it not only simplifies calculations but also helps in removing radicals from the denominator.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing them to their simplest form. This process is made easier when you rationalize denominators and use the difference of squares. Here are some steps to simplify expressions like the one given in the exercise:

• Multiply by the Conjugate: As shown in the exercise, multiply by a conjugate to eliminate radicals from the denominator.
• Expand and Simplify: Expand the expression in the numerator, distributing any multiplied terms, and simplify.
• Final Marvel: Simplified expressions often result in cleaner, more manageable fractions without square roots in the denominator.

For example, after multiplying and expanding, our numerator becomes \(10\sqrt{5} - 15\sqrt{7}\). Dividing by the simplified denominator \(-43\), results in \[-\frac{10\sqrt{5}}{43} + \frac{15\sqrt{7}}{43}\].This illustrates how, by recognizing and applying the correct mathematical identities and processes, complex-looking radical expressions can be simplified for easier understanding and further manipulation.