Problem 54 - College Algebra Essentials

Question

In Exercises \(45-54,\) rationalize the denominator. $$\frac{11}{\sqrt{7}-\sqrt{3}}$$

Step-by-Step Solution

Answer

The simplified fraction with a rationalized denominator is \(\frac{11\sqrt{7}+11\sqrt{3}}{4}\).

Step 1: Identify the conjugate of the denominator

The conjugate of the denominator \(\sqrt{7}-\sqrt{3}\) is \(\sqrt{7}+\sqrt{3}\).

Step 2: Multiply the fraction by the conjugate

Multiply the fraction \(\frac{11}{\sqrt{7}-\sqrt{3}}\) by \(\frac{\sqrt{7}+\sqrt{3}}{\sqrt{7}+\sqrt{3}}\). This results in \(\frac{11(\sqrt{7}+\sqrt{3})}{(\sqrt{7}-\sqrt{3})(\sqrt{7}+\sqrt{3})}\).

Step 3: Simplify the expression

Multiplying out the brackets, the fraction becomes \(\frac{11\sqrt{7}+11\sqrt{3}}{7-3}\), which simplifies to \(\frac{11\sqrt{7}+11\sqrt{3}}{4}\).

Key Concepts

Understanding Conjugates

When dealing with irrational numbers in the denominator of a fraction, we utilize conjugates to make the expressions rational.
Conjugates are pairs of expressions formed by changing the sign between two terms. For example, if we have \( \sqrt{7} - \sqrt{3} \), its conjugate will be \( \sqrt{7} + \sqrt{3} \).

Multiplying an expression by its conjugate helps to eliminate the radical parts when it's in the denominator:

• This is because the product of two conjugates, \((a + b)(a - b)\), follows the pattern of a difference of squares: \(a^2 - b^2\).
• For \(\sqrt{7} - \sqrt{3}\), the product with its conjugate \(\sqrt{7} + \sqrt{3}\) results in \(7 - 3 = 4\).

The goal is to replace the radical in the denominator with a simple number.

Working with Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials or can include variables and radicals.
They can often be tricky to manipulate due to these elements.

Key points when working with them include:

• Ensuring that you multiply the numerator and denominator by the same value to keep the expression equivalent.
• Simplifying where possible to make the expression more manageable.

In our exercise, we use the conjugate to help rationalize the denominator, converting \( \frac{11}{\sqrt{7} - \sqrt{3}} \) into a more straightforward form.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing them to their simplest form.
This can include multiplying and dividing radicals, as well as performing operations like addition and subtraction where applicable.

Steps for simplifying include:

• Look for opportunities to use properties of radicals, like \((\sqrt{a})^2 = a\).
• Combine like terms, such as \(11\sqrt{7} + 11\sqrt{3}\) becoming part of a whole expression.

In our case, to simplify \( \frac{11(\sqrt{7} + \sqrt{3})}{4} \), we treat the radical parts separately and ensure the entire fraction is in its simplest form.