Problem 99 - Elementary and Intermediate Algebra

Question

Rationalize each numerator. All variables represent positive real numbers. $$ \frac{\sqrt{x}+3}{x} $$

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{x - 9}{x(\sqrt{x} - 3)} \).

Step 1: Identify the Conjugate

To rationalize the numerator, we need to multiply both the numerator and the denominator by the conjugate of the numerator. The conjugate of \( \sqrt{x} + 3 \) is \( \sqrt{x} - 3 \).

Step 2: Multiply Numerator and Denominator

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

Step 3: Simplify the Numerator

Use the difference of squares formula to simplify the numerator. Recall that \((a + b)(a - b) = a^2 - b^2\). Thus,\[(\sqrt{x} + 3)(\sqrt{x} - 3) = (\sqrt{x})^2 - 3^2 = x - 9\]

Step 4: Write the Rationalized Expression

After substituting the simplified numerator, we have the final rationalized expression:\[\frac{x - 9}{x(\sqrt{x} - 3)}\]

Key Concepts

Conjugates in Algebra

Conjugates in algebra are a key tool when dealing with expressions involving square roots. The conjugate of an expression like \( \sqrt{x} + 3 \) is \( \sqrt{x} - 3 \). Notice how the sign in the middle changes from positive to negative. This simple change is incredibly useful.

This concept of using a conjugate helps in rationalizing expressions. Rationalizing means removing the square root from the numerator or denominator, making expressions easier to handle. By multiplying by the conjugate, we aim to simplify complex fractions.

Using the conjugate is a strategic move in algebraic operations. It allows expressions to be rewritten in a form that is often easier to evaluate and understand, especially when tackling more challenging algebraic problems later on. This technique is not limited to binomials but can be extended to complex numbers as well. Remember, the main idea is to create a multiplication that results in the cancellation of the square root, leading us to cleaner and more simplified expressions.

Difference of Squares

The difference of squares is a crucial algebraic identity that frequently appears in the process of rationalizing numerators. This identity states that for any two terms \( a \) and \( b \), the product \((a + b)(a - b)\) simplifies to \( a^2 - b^2 \).

In the context of our original problem, we used the difference of squares to simplify the product of the conjugate and the original numerator. Applying the formula to \((\sqrt{x} + 3)(\sqrt{x} - 3)\), we get:

• Set \( a = \sqrt{x} \) and \( b = 3 \).
• Then \( a^2 = (\sqrt{x})^2 = x \).
• Also, \( b^2 = 3^2 = 9 \).

Plug these into \( a^2 - b^2 \) to get \( x - 9 \).

This algebraic trick of recognizing and applying the difference of squares greatly simplifies expressions. It highlights the power of mathematical identities in transforming complex algebraic expressions into more manageable forms.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of making them easier to understand and solve. This can involve several techniques, including using identities like the difference of squares, distributing terms, and collecting like terms.

When rationalizing the numerator in our exercise, we didn't just stop at multiplying by the conjugate. We went on to simplify the resulting expression, which is fundamental to mastering algebra.

After using the conjugate, simplifying the numerator leverages the difference of squares. We broke down the expression \((\sqrt{x} + 3)(\sqrt{x} - 3)\) into its simplest form, \(x - 9\). This step is crucial as it transforms the problem into something that can be handled more easily.

Remember:

• Always look for opportunities to apply algebraic identities, as they make expressions simpler.
• Clear away square roots in the numerator or denominator through rationalization when needed.
• Simplifying means finding the most efficient form of an expression for easier use in computations or further algebraic manipulations.

By practicing these skills, handling complex algebraic tasks becomes more intuitive and straightforward.