Problem 13 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{5 \sqrt{5}}{15 \sqrt{2}}\)

Step-by-Step Solution

Answer

The simplified form of \(\frac{5 \sqrt{5}}{15 \sqrt{2}}\) is \(\frac{\sqrt{10}}{6}\).

Step 1: Identify the Reason to Rationalize

To rationalize the denominator of the given fraction \(\frac{5 \sqrt{5}}{15 \sqrt{2}}\), we need to remove the square root from the denominator to make it a rational number.

Step 2: Multiply by a Conjugate

Since the denominator is \(15 \sqrt{2}\), multiply both the numerator and the denominator by \(\sqrt{2}\) to eliminate the square root in the denominator. Our expression becomes \(\frac{5 \sqrt{5} \cdot \sqrt{2}}{15 \sqrt{2} \cdot \sqrt{2}}\).

Step 3: Simplify the Denominator

Calculate \(\sqrt{2} \cdot \sqrt{2} = 2\), so the denominator becomes \(15 \times 2 = 30\).

Step 4: Simplify the Numerator

Multiply \(\sqrt{5} \cdot \sqrt{2}\) in the numerator, resulting in \(\sqrt{10}\). This gives \(5 \sqrt{10}\) in the numerator.

Step 5: Write the New Fraction

The expression is now \(\frac{5 \sqrt{10}}{30}\).

Step 6: Simplify the Fraction Completely

Reduce \(\frac{5}{30}\) by dividing both the numerator and denominator by their greatest common divisor, which is 5. Therefore, \(\frac{5}{30} = \frac{1}{6}\). The fraction simplifies to \(\frac{\sqrt{10}}{6}\).

Key Concepts

Square Roots

A square root represents a number that, when multiplied by itself, gives the original number. For instance, the square root of 4 is 2 because

• 2 multiplied by itself equals 4.

Square roots are common in mathematical expressions, often denoted by a radical sign \( \sqrt{} \). They are vital in many mathematical operations.
When dealing with fractions, square roots might appear in the numerator, denominator, or both. To rationalize an expression, we often aim to eliminate the square root from the denominator, making it a rational number.

In our example exercise, the denominator is \(15\sqrt{2}\). By multiplying by \( \sqrt{2} \), we turn the denominator into 30, a rational number without square roots. This step not only simplifies computation but is also a standard practice in mathematics.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means expressing the fraction such that the numerator and denominator have no common factors other than 1. This simplification helps to make numbers more manageable and easier to interpret.

When we talk about simplifying, it involves:

• Identifying and dividing both the numerator and denominator by their greatest common divisor (GCD).
• For instance, if we have a fraction like \( \frac{5}{30} \), the GCD is 5.
• Dividing both the numerator and denominator by 5 results in \( \frac{1}{6} \).

In mathematical expressions containing square roots, after rationalizing, we also aim to simplify the fraction completely.
For our step-by-step solution, this was the final step, turning \( \frac{5 \sqrt{10}}{30} \) into \( \frac{\sqrt{10}}{6} \). This process ensures the fraction is easier to work with in any subsequent mathematical operations.

Mathematical Expressions

Mathematical expressions are combinations of numbers, operators, and symbols that represent a value. They can include a variety of elements, such as variables and constants, mixed with operational signs like addition, subtraction, multiplication, and division.

Handling and transforming these expressions is key in mathematics. This often involves simplifying expressions for ease of use. When expressions have square roots, like in our problem, rationalizing helps us transform them into a more usable form.

To deal with expressions involving square roots:

• Identify sections of the expression to transform, such as the denominator in a fraction.
• Use multiplication or division to simplify the expression or make it rational.
• Ensure to follow mathematical principles to maintain balance in the expression.

In our example, multiplying \( \sqrt{2} \) to both numerator and denominator was a crucial step in rationalizing the expression \( \frac{5 \sqrt{5}}{15 \sqrt{2}} \), demonstrating manipulation of mathematical expressions for simplification.