Problem 50 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The expression in simplest radical form is \(\frac{\sqrt{21}}{7}\).

Step 1: Understand the Problem

We need to simplify the expression \(\frac{\sqrt{3}}{\sqrt{7}}\) by eliminating the radical in the denominator.

Step 2: Rationalize the Denominator

To rationalize the denominator, we multiply both the numerator and the denominator by \(\sqrt{7}\), which is the conjugate of \(\sqrt{7}\). This gives us: \[ \frac{\sqrt{3} \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} \]

Step 3: Simplify the Expression

Now, apply the property of square roots, which states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Thus, we have: \[ \frac{\sqrt{21}}{\sqrt{49}} \] Simplifying the denominator further, we know that \(\sqrt{49} = 7\). Thus, we have: \[ \frac{\sqrt{21}}{7} \] This is the simplest radical form.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate radicals from the bottom of a fraction.
This process makes it easier to work with and compare numbers, especially when dealing with square roots. In essence, we aim to remove any irrational numbers from the denominator by converting them into a rational number format.

• For any expression like \( \frac{\sqrt{a}}{\sqrt{b}} \), our primary goal is to remove \( \sqrt{b} \) from the denominator.
• We achieve this by multiplying both the numerator and the denominator by \( \sqrt{b} \), which doesn't change the value of the fraction because you're multiplying by 1 (\( \sqrt{b}/\sqrt{b} = 1 \)).

After this multiplication, the denominator becomes \( b \) because \( \sqrt{b} \times \sqrt{b} = b \). The overall fraction is simpler, making computations and comparisons more straightforward.

Properties of Square Roots

Understanding square roots is key to simplifying radical expressions. Square roots have specific properties that help in their manipulation.
These properties can also simplify expressions by combining or separating them.

• The product property states: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This helps when multiplying radicals to simplify an expression.
• The division property asserts: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). This allows expressions to be simplified into a single square root if the division is possible without breaking out to decimals.

In our example, when simplifying \( \frac{\sqrt{21}}{\sqrt{49}} \), knowing \( \sqrt{49} = 7 \) helps us write the simplest radical form. Such properties make dealing with square roots more manageable.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest form.
This concept focuses on ensuring that no perfect square factors remain under a square root, which makes the expression easier to read and use.

• Breaking down the number or expression under the radical into its prime factors helps in identifying any factors that can be "taken out" of the radical.
• If a factor appears in pairs under the square root, it can be "taken out" of the square root; for example, \( \sqrt{4x^2} = 2x \).

In the solution \( \frac{\sqrt{21}}{7} \), \( 21 \) does not have perfect square factors other than 1, so \( \sqrt{21} \) is as simple as the expression can get. This attention to simplicity helps in further mathematical computations and clarifies expressions.