Problem 95 - Intermediate Algebra

Question

Determine the smallest number both the numerator and denominator should be multiplied by to rationalize the denominator of the radical expression. See the Concept Check in this section. When rationalizing the denominator of \(\frac{\sqrt{5}}{\sqrt{7}}\) explain why both the numerator and the denominator must be multiplied by \(\sqrt{7}\)

Step-by-Step Solution

Answer

Multiply both by \( \sqrt{7} \) to rationalize the denominator.

Step 1: Understand the Concept

When rationalizing the denominator of a fraction that involves a radical, the goal is to eliminate the square root in the denominator. This is done by multiplying both the numerator and the denominator by the radical in the denominator or by its conjugate, if it is more complex.

Step 2: Identify the Denominator

In the expression \( \frac{\sqrt{5}}{\sqrt{7}} \), the denominator is \( \sqrt{7} \). This is the term we want to eliminate from the denominator.

Step 3: Multiply by the Denominator's Radical

To eliminate the square root of 7 from the denominator, multiply both the numerator and the denominator by \( \sqrt{7} \). This results in a new fraction: \( \frac{\sqrt{5} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} \).

Step 4: Simplify the Denominator

The denominator \( \sqrt{7} \cdot \sqrt{7} \) simplifies to \( 7 \), since the product of a square root by itself is the radicand, making our expression \( \frac{\sqrt{35}}{7} \).

Step 5: Review the Completed Expression

The initial fraction \( \frac{\sqrt{5}}{\sqrt{7}} \) has been rationalized to \( \frac{\sqrt{35}}{7} \) by removing the square root from the denominator.

Key Concepts

Understanding Radical Expressions

Radical expressions are expressions that contain a radical symbol, like a square root, cube root, or other roots. The most common radical is the square root, represented by the symbol \( \sqrt{} \). Radical expressions can be found in many mathematical situations and can sometimes seem complex.
However, when working with these expressions, particularly in fractions, it is important to simplify in order to make calculations easy.

• If you see a square root symbol, understand that it is asking for the number which, when multiplied by itself, gives the original number under the root.
• For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).

When radicals appear in the denominator of a fraction, mathematicians often prefer to "rationalize" them by eliminating the radical.
This is because fractions with radicals in the denominator can be more complex to handle and compare.

Simplifying Radicals Effectively

Simplifying radicals involves breaking down the expression to its simplest form. This process makes it easier to work and perform operations with radicals.
Simplifying radicals often requires breaking the number under the root into its prime factors and taking out pairs.

• Identify the radicand, the number under the root, and find its prime factorization.
• Look for pairs of factors because a pair can be brought out of the radical as a single unit.

For instance, let's consider \( \sqrt{18} \).
When we break down 18, we find it is made of \( 2 \times 3 \times 3 \). Since we have a pair of 3's, \( \sqrt{18} \) can be simplified as \( 3\sqrt{2} \) by taking the 3 out of the radical. Simplifying radicals makes computations and understanding expressions much easier.

Performing Fraction Operations

Fractions can sometimes involve radicals, which adds an extra layer of complexity.
When you encounter a fraction with a radical in the denominator, as in the case of \( \frac{\sqrt{5}}{\sqrt{7}} \), you can use the method of rationalizing the denominator.

• Identify the radical in the denominator.
• Multiply both the numerator and the denominator by this radical.

This process turns the denominator into a rational number, simplifying the whole fraction. In the given example, multiplying both numerator and denominator by \( \sqrt{7} \) gives \( \frac{\sqrt{5} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} \).
The denominator becomes 7, which is rational, and the fraction is now \( \frac{\sqrt{35}}{7} \). The process of rationalizing by multiplying by the radical eliminates the square root from the denominator, making computations more straightforward.
Practicing these steps with different numbers helps in mastering the simplification and operations of fractions and radicals.