Problem 57 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{2 \sqrt{3}}{\sqrt{7}}\)

Step-by-Step Solution

Answer

\(\frac{2 \sqrt{21}}{7}\)

Step 1: Understand the Problem

We need to change the expression \(\frac{2 \sqrt{3}}{\sqrt{7}}\) into its simplest radical form. This involves getting rid of the radical in the denominator.

Step 2: Identify the Method

To simplify the expression, we can use the process called 'rationalizing the denominator,' which involves multiplying the expression by a form of 1 that will remove the square root from the denominator.

Step 3: Multiply by the Conjugate

Multiply both the numerator and the denominator by \(\sqrt{7}\). This gives us \(\frac{2 \sqrt{3} \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}}\).

Step 4: Simplify the Expression

Multiply the terms: the numerator becomes \(2 \sqrt{21}\) because \(\sqrt{3} \times \sqrt{7} = \sqrt{21}\), and the denominator becomes 7, as \(\sqrt{7} \times \sqrt{7} = 7\).

Step 5: Write the Simplified Form

So the simplest radical form of \(\frac{2 \sqrt{3}}{\sqrt{7}}\) is \(\frac{2 \sqrt{21}}{7}\).

Key Concepts

Rationalizing the Denominator

When we say rationalizing the denominator, we're talking about the process of removing any square roots or radicals from the bottom of a fraction. In math, it's cleaner and often preferred to have a rational number in the denominator.

Why is this important? Well, operations with fractions are generally easier when the denominator does not contain a radical. Let's see how it works:

• We have a fraction like \(\frac{2 \sqrt{3}}{\sqrt{7}}\).
• The goal is to remove \(\sqrt{7}\) from the bottom.

To do this, we multiply both the top and bottom of the fraction by \(\sqrt{7}\), which is a form of 1 that keeps the fraction balanced.

This is because multiplying by 1 doesn’t change the value of a number. The outcome is a denominator of 7, since \(\sqrt{7} \times \sqrt{7} = 7\), making the denominator rational.

Simplest Radical Form

An expression is in its simplest radical form when there's no perfect square factor remaining in the radicand (the number under the radical), and no radical is left in the denominator.

For our expression \(\frac{2 \sqrt{3}}{\sqrt{7}}\), the final simplest form is \(\frac{2 \sqrt{21}}{7}\). Here's how we achieved it:

• Multiply the radicals: \(\sqrt{3} \times \sqrt{7} = \sqrt{21}\), a process known as multiplying radicals.
• Check if the number 21 has any perfect square factors. It doesn’t (since 21 is \(3 \times 7\)), so \(\sqrt{21}\) is already simplified.

The expression \(\frac{2 \sqrt{21}}{7}\) doesn't have any radicals in the denominator and the numerator's radical is minimized, achieving the simplest form.

Algebraic Expressions

Algebraic expressions can sometimes involve radicals, especially when simplifying or solving equations. Understanding these concepts can make algebra more approachable and easier to manage.

Algebraic expressions consist of terms that can include:

• Constants (e.g., numbers)
• Variables (e.g., \(x\), \(y\))
• Radicals, which are used to represent roots (e.g., \(\sqrt{x}\))

When working with radicals in algebraic expressions, it helps to break down the expression to make it manageable. In our earlier expression \(\frac{2 \sqrt{3}}{\sqrt{7}}\), the variables were not present, but the process of simplifying remained similar.

Radicals require special attention to keep calculations precise, especially when rationalizing the denominator or simplifying to the simplest radical form. Once you grasp these principles, algebraic expressions with radicals become tasks you can confidently tackle.