Problem 68 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The rationalized form is \( \frac{3\sqrt{x} - 21}{x - 49} \).

Step 1: Identify the Conjugate

To rationalize the denominator, we start by identifying the conjugate of the denominator. Given the denominator is \( \sqrt{x} + 7 \), the conjugate is \( \sqrt{x} - 7 \).

Step 2: Multiply the Numerator and Denominator by the Conjugate

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

Step 3: Simplify the Denominator Using Difference of Squares

Use the formula for the difference of squares: \( (a+b)(a-b) = a^2 - b^2 \). In this case, \( a^2 = (\sqrt{x})^2 = x \) and \( b^2 = 7^2 = 49 \). So, the denominator simplifies to:\[x - 49.\]

Step 4: Expand and Simplify the Numerator

Expand the numerator: \[3(\sqrt{x}-7) = 3\sqrt{x} - 21.\]

Step 5: Combine and Simplify the Fraction

Combine the simplified numerator and denominator:\[\frac{3\sqrt{x} - 21}{x - 49}.\]This is the expression with a rationalized denominator.

Key Concepts

Difference of Squares

The difference of squares is a significant algebraic concept that simplifies expressions involving squares. It is based on the formula \((a+b)(a-b) = a^2 - b^2\).In simpler terms, when you multiply a sum by a difference, you get the difference between the squares of these numbers.

For instance, if you have the terms \(a = \sqrt{x}\) and \(b = 7\), applying the formula gives \((\sqrt{x} + 7)(\sqrt{x} - 7) = (\sqrt{x})^2 - 7^2 = x - 49\).This technique helps remove the square roots from the denominator, making the expression easier to handle in further calculations.

Recognizing opportunities to use the difference of squares can simplify solving complex algebraic problems. Memorizing the basic formula \((a+b)(a-b) = a^2 - b^2\) will empower you to tackle challenging expressions with confidence.

Simplifying Fractions

Simplifying fractions is all about making an expression easier to understand by reducing its components as much as possible. After rationalizing the denominator, the goal is to simplify anything that remains. Let's break down the process.

• First, perform any multi-step arithmetic operations like expanding a multiplication or distribution. For example, if you have a term like \(3(\sqrt{x}-7)\), distribute the 3, resulting in \(3\sqrt{x} - 21\).
• Next, look at both the numerator and the denominator to see if they share any common factors that can be canceled out. This isn't always possible, but it's often the final step in simplifying a fraction.

In our expression \(\frac{3\sqrt{x} - 21}{x - 49}\), we find the terms cannot be simplified further because they do not have common factors. Simplifying fractions ensures clarity and reduces the complexity of the problem.

Conjugate Pairs

Conjugate pairs play a vital role in algebra, particularly in rationalizing denominators. A conjugate pair occurs when you have two expressions of the form \(a + b\) and \(a - b\). These pairs help eliminate radicals from the denominator.

In this exercise, the denominator is \(\sqrt{x} + 7\). Its conjugate is \(\sqrt{x} - 7\). By multiplying the original fraction by the conjugate over itself, \(\frac{3}{\sqrt{x}+7}\times \frac{\sqrt{x}-7}{\sqrt{x}-7}\),we maintain the fraction's original value. The product \((\sqrt{x}+7)(\sqrt{x}-7)\) results in a difference of squares, \(x - 49\), which is easier to compute.

Using conjugate pairs is an efficient strategy in algebra to simplify seemingly complex fractions and make them more manageable.