Problem 67 - Algebra for College Students

Question

Rationalize the denominator and simplify. All variables represent positive real numbers. \(\frac{2}{\sqrt{x}+4}\)

Step-by-Step Solution

Answer

The rationalized expression is \(\frac{2\sqrt{x} - 8}{x-16}\).

Step 1: Understanding the Problem

We need to rationalize the denominator of the fraction \( \frac{2}{\sqrt{x}+4} \). This involves eliminating the square root from the denominator.

Step 2: Identify the Conjugate

The conjugate of \( \sqrt{x}+4 \) is \( \sqrt{x}-4 \). Multiplying by this conjugate will help eliminate the square root in the denominator.

Step 3: Multiply Numerator and Denominator by the Conjugate

To rationalize, multiply both the numerator and the denominator by the conjugate: \( \frac{2}{\sqrt{x}+4} \times \frac{\sqrt{x}-4}{\sqrt{x}-4} \).

Step 4: Distribute in Numerator

Multiply the numerator: \( 2 \times (\sqrt{x}-4) = 2\sqrt{x} - 8 \).

Step 5: Use the Difference of Squares in Denominator

The denominator becomes \((\sqrt{x}+4)(\sqrt{x}-4) = x - 16\) because \((a+b)(a-b) = a^2 - b^2\). Here, \( a = \sqrt{x} \) and \( b = 4 \).

Step 6: Write the Rationalized Expression

The expression with the rationalized denominator is \( \frac{2\sqrt{x} - 8}{x-16} \).

Key Concepts

Conjugate: A Key Tool in Rationalizing Denominators

The concept of a conjugate is crucial when you're aiming to rationalize the denominator of an expression that includes a square root. A conjugate is essentially formed by changing the sign between two terms. For instance, the conjugate of \( \sqrt{x} + 4 \) is \( \sqrt{x} - 4 \). By multiplying a fraction by the conjugate of its denominator, we eliminate the square root, making the expression easier to manage.
Remember:

• To find the conjugate, only change the sign between the two terms.
• Conjugates assist in getting rid of pesky radicals in the denominator.

In our example, multiplying by \( \frac{\sqrt{x}-4}{\sqrt{x}-4} \) ensures that the square root is removed from the denominator. This approach is valid because it does not change the value of the fraction; multiplying by\( 1 \) keeps the expression equivalent.

Simplifying Expressions: Making Math Manageable

Simplifying expressions is all about making the math easier to handle. By reducing expressions to their simplest form, it becomes easier to perform further calculations or interpretations. In the original problem, after multiplying by the conjugate, you need to simplify the result to maintain clarity and accuracy.
Here’s how you simplify effectively:

• First, distribute any terms in the numerator.
• Simplify any potential products or combine like terms.

In our case, multiplying in the numerator: \( 2 \times (\sqrt{x} - 4) \) results in \( 2\sqrt{x} - 8 \). Simplifying shouldn’t alter the value of the original expression, it just makes it clearer by removing unnecessary complexity.

Difference of Squares: A Handy Multiplication Trick

When faced with expressions like \((a + b)(a - b)\), the difference of squares is a powerful technique to simplify them. This is a formula where you subtract the square of one term from the square of another: \((a+b)(a-b) = a^2 - b^2\).
The difference of squares has.

• helps in transforming the original expression.
• involves recognizable patterns, making calculation straightforward.

In our exercise, the expression \((\sqrt{x} + 4)(\sqrt{x} - 4)\) converts into \(x - 16\) using this formula, where \(a = \sqrt{x}\) and \(b = 4\). This new form is much easier to work with and ensures no radicals remain in the denominator.