Problem 70 - Algebra A Combined Function

Question

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{5+\sqrt{2}}{\sqrt{2 x}}\)

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{23}{5\sqrt{2x} - 2\sqrt{x}} \).

Step 1: Understanding the Problem

We are given the expression \(\frac{5+\sqrt{2}}{\sqrt{2x}}\) and are asked to rationalize the numerator. This means we need to eliminate any radicals in the numerator of the expression.

Step 2: Express Idea of Rationalization

To rationalize the numerator \(5+\sqrt{2}\), we multiply both the numerator and the denominator by the conjugate of the numerator. The conjugate of \(5+\sqrt{2}\) is \(5-\sqrt{2}\).

Step 3: Perform Multiplication

Multiply the numerator \((5+\sqrt{2})(5-\sqrt{2})\) using the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\). Here \(a=5\) and \(b=\sqrt{2}\). So, the numerator becomes \(5^2 - (\sqrt{2})^2 = 25 - 2 = 23\).

Step 4: Simplify Denominator

Multiply the denominator \(\sqrt{2x}\) by \(5-\sqrt{2}\). The denominator becomes \((\sqrt{2x})(5-\sqrt{2}) = 5\sqrt{2x} - \sqrt{4x} = 5\sqrt{2x} - 2\sqrt{x}\).

Step 5: Combine and Finalize Expression

After performing the multiplications, the expression is: \[\frac{23}{5\sqrt{2x} - 2\sqrt{x}}\]. This is the final expression with a rationalized numerator.

Key Concepts

Conjugate Pairs

When dealing with expressions involving radicals, conjugate pairs are incredibly useful. A conjugate pair consists of two expressions that are identical except for the sign between their terms. In our exercise, the expression was \(5+\sqrt{2}\), and its conjugate pair is \(5-\sqrt{2}\). By using these pairs, we can eliminate radicals from certain parts of our expression. This happens because when you multiply conjugates, the middle terms cancel out, leaving a simpler result without radicals.

• To find a conjugate, just change the sign between the terms.
• Multiplying conjugates uses the "Difference of Squares" formula.

Understanding conjugates helps simplify expressions and make them easier to work with, especially when it comes to rationalizing numerators or denominators in a fraction.

Square Roots

The square root operation is a fundamental component of working with radicals. The square root of a number \(a\) is a value \(b\) such that \(b^2 = a\). In rational expressions like the one we solved, square roots appear in numerators or denominators and can often complicate calculations. One of the main goals when handling such expressions is to eliminate these roots from undesirable locations.

• Square roots can often be simplified by identifying perfect squares.
• Removing square roots from the numerator or denominator makes the expression more manageable.

Using conjugate pairs is one effective strategy for dealing with square roots, especially when they are part of sums or differences.

Difference of Squares

The difference of squares is an algebraic formula that states \((a+b)(a-b) = a^2 - b^2\). It's a key tool for simplifying expressions, particularly when it comes to multiplying conjugates. In the step-by-step solution process for our expression, we multiplied \(5+\sqrt{2}\) by its conjugate \(5-\sqrt{2}\), applying the difference of squares formula.

• This formula helps cancel out terms that involve radicals.
• It results in a difference of two squared terms, simplifying the expression.

Knowing how to apply the difference of squares efficiently is crucial for simplifying expressions and has broad applications in algebra.

Radicals

Radicals, including square roots, are expressions that contain a root symbol. They represent the inverse operation of raising a number to a power. Working with radicals often requires simplifying them for easier mathematical manipulation. In our example, we had to rationalize the numerator of a radical expression.

• Rationalization involves removing radicals from the numerator or denominator.
• Conjugate pairs are a common method used to achieve this.

Mastering radicals includes understanding how to work with them within fractions and knowing when they can be simplified or completely eliminated from an expression.