Problem 87 - A Graphical Approach to College Algebra

Question

Rationalize the denominator of each radical expression. Assume that all variables represent nonnegative real numbers and that no denominators are \(0 .\) $$\frac{p}{\sqrt{p}+2}$$

Step-by-Step Solution

Answer

\( \frac{p\sqrt{p} - 2p}{p - 4} \) is the rationalized expression.

Step 1: Identifying the Conjugate

To rationalize the denominator, identify the conjugate of the denominator. The original denominator is \( \sqrt{p} + 2 \). The conjugate of this expression is \( \sqrt{p} - 2 \).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate. The expression becomes:\[\frac{p}{\sqrt{p} + 2} \times \frac{\sqrt{p} - 2}{\sqrt{p} - 2} = \frac{p(\sqrt{p} - 2)}{(\sqrt{p} + 2)(\sqrt{p} - 2)}.\]

Step 3: Multiply the Denominator

Use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) on the denominator:\[(\sqrt{p})^2 - 2^2 = p - 4.\]

Step 4: Multiply the Numerator

Distribute \(p\) in the numerator:\[p \cdot (\sqrt{p} - 2) = p\sqrt{p} - 2p.\]

Step 5: Simplify the Expression

Combine the results from the previous steps to write the final rationalized expression:\[\frac{p\sqrt{p} - 2p}{p - 4}.\]

Key Concepts

Conjugates

The concept of conjugates is pivotal when it comes to rationalizing the denominator of a radical expression. Conjugates are pairs of expressions involving radicals that are identical except for the sign between two terms. For instance, if you have the expression \(a + b\), its conjugate would be \(a - b\). In the original exercise, we had the denominator \(\sqrt{p} + 2\), and its conjugate was \(\sqrt{p} - 2\).

• Identifying and using conjugates helps simplify expressions with radicals in the denominator.
• Multiplying expressions by their conjugate takes advantage of a special formula: the difference of squares.

Each time you multiply conjugates, the result is a rational expression, free of radicals, making it simpler to handle. This is an especially useful trick in simplifying complex algebraic fractions.

Radical Expressions

Radical expressions often contain square roots or other roots, and dealing with them can sometimes be tricky. These expressions can appear in various forms, and our main goal in rationalizing the denominator is to remove the radicals from the denominator.

• Square roots like \(\sqrt{p}\) are the most common radical expressions one encounters in these problems.
• When combined with other terms, the goal is to rationalize the fraction for easier mathematical manipulation.

To rationalize, you multiply both the numerator and the denominator by the conjugate that contains the same radical expression. Let's see what happened in the example: we dealt with the expression \(\frac{p}{\sqrt{p}+2}\) by multiplying both parts by its conjugate \(\sqrt{p} - 2\), successfully moving the radical from the denominator.

Difference of Squares

The difference of squares is a powerful algebraic formula used in manipulating expressions, particularly useful for rationalizing denominators. Given two terms, \(a\) and \(b\), the formula states: \[ (a+b)(a-b) = a^2 - b^2 \] This formula is directly applied when multiplying the original denominator by its conjugate. In our example, we have

• \((\sqrt{p} + 2)\) and \( (\sqrt{p} - 2)\), leading to \(a = \sqrt{p}\) and \(b = 2\).
• Applying the formula results in \((\sqrt{p})^2 - 2^2 \), which simplifies to \(p - 4\).

Through this process, we effectively removed the radical terms from the denominator, showing how the difference of squares formula essentially transforms the expression into a more manageable form. This technique is a staple in algebra when dealing with rational expressions and helps uphold the structure of mathematical operations.