Problem 96 - 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 numerator of \(\frac{\sqrt{5}}{\sqrt{7}}\) explain why both

Step-by-Step Solution

Answer

Multiply both by \( \sqrt{7} \), resulting in \( \frac{\sqrt{35}}{7} \).

Step 1: Identify the problem

We need to rationalize the denominator of the expression \( \frac{\sqrt{5}}{\sqrt{7}} \). This means removing the radical from the denominator.

Step 2: Multiply by the conjugate

To eliminate the radical in the denominator, multiply both the numerator and the denominator by \( \sqrt{7} \), which is the conjugate of \( \sqrt{7} \). This step results in the expression being \( \frac{\sqrt{5} \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} \).

Step 3: Simplify the expression

Multiply the expressions: The numerator becomes \( \sqrt{35} \) since \( \sqrt{5} \times \sqrt{7} = \sqrt{35} \). The denominator becomes 7 because \( \sqrt{7} \times \sqrt{7} = 7 \). The simplified expression is \( \frac{\sqrt{35}}{7} \).

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that involve roots, such as square roots or cube roots. These expressions are common in algebra and often require manipulation to simplify or solve equations. For example, in the expression \( \frac{\sqrt{5}}{\sqrt{7}} \), both the numerator and the denominator are under a square root, which makes them radical expressions.

To work with radical expressions effectively, it's important to understand the properties of roots. The square root \( \sqrt{a} \) represents a number that, when multiplied by itself, yields \( a \). When dealing with these expressions, you might need to simplify or rationalize them, which often involves other concepts like conjugates or numerical factors.

Conjugate Multiplication

Conjugate multiplication is a technique used to rationalize denominators, especially when dealing with square roots or other irrational numbers in the denominator. This method involves multiplying both the numerator and the denominator by the conjugate of the denominator.

• The conjugate of \( \sqrt{a} + b \) is \( \sqrt{a} - b \), and vice versa.
• In the expression \( \frac{\sqrt{5}}{\sqrt{7}} \), the conjugate of \( \sqrt{7} \) is itself, \( \sqrt{7} \), because it doesn't have a binomial form.

By multiplying both the numerator and denominator by \( \sqrt{7} \), the denominator becomes a rational number. This works because \( \sqrt{7} \times \sqrt{7} \) equals 7, which is a rational number, effectively removing the radical.

This technique is very powerful, as it allows us to convert an otherwise difficult-to-interpret expression into a more manageable and straightforward form.

Simplification of Fractions

Simplifying fractions involves reducing them to their simplest form, which makes evaluating and working with them easier. This often involves finding the greatest common divisor or eliminating radicals from the denominator. When dealing with expressions like \( \frac{\sqrt{35}}{7} \), simplification can make the expression easier to understand and use.

• Ensure all radicals are removed from the denominator.
• Simplify the radicals in the numerator, if possible.

In this case, after multiplying by the conjugate \( \sqrt{7} \), we arrived at \( \frac{\sqrt{35}}{7} \). Since the denominator is now a rational number, the expression is considered simplified. This process not only clarifies the expression but also makes it more useful for further algebraic manipulation.

Understanding how to simplify fractions ensures you can handle more complex mathematical problems with ease.