Problem 48 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{\frac{7}{12}}\)

Step-by-Step Solution

Answer

The simplest radical form of \( \sqrt{\frac{7}{12}} \) is \( \frac{\sqrt{21}}{6} \).

Step 1: Write Down the Original Expression

The expression we need to simplify is \( \sqrt{\frac{7}{12}} \).

Step 2: Simplify the Expression Under the Radical

The expression under the radical is a fraction, \( \frac{7}{12} \). This can be expressed as the square root of the numerator over the square root of the denominator: \( \frac{\sqrt{7}}{\sqrt{12}} \).

Step 3: Simplify the Denominator

The square root of the denominator, \( \sqrt{12} \), can be simplified. Notice that \( 12 = 4 \times 3 \). Thus, \( \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \).

Step 4: Substitute the Simplified Denominator

Replace \( \sqrt{12} \) with \( 2 \sqrt{3} \) in the expression: \( \frac{\sqrt{7}}{2\sqrt{3}} \).

Step 5: Rationalize the Denominator

To simplify further, multiply the numerator and the denominator by \( \sqrt{3} \), which is the conjugate of the denominator: \( \frac{\sqrt{7} \cdot \sqrt{3}}{2\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{21}}{6} \).

Step 6: Write the Simplified Expression

After rationalizing the denominator and simplifying, the expression is \( \frac{\sqrt{21}}{6} \).

Key Concepts

Rationalizing Denominators

Rationalizing denominators is a process used in algebra to eliminate radicals from the denominator of a fraction. This helps simplify expressions and makes them easier to work with. When you have a radical in the denominator, like in the expression \(\frac{\sqrt{7}}{2\sqrt{3}}\), you multiply both the numerator and the denominator by a number that will remove the radical from the denominator. In this case, multiplying by \(\sqrt{3}\) works because \(2\sqrt{3} \cdot \sqrt{3} = 6\), a rational number.

• Eliminate radicals in the denominator by multiplying with the appropriate value.
• Convert the denominator to a rational number for simplicity.
• Keep your solution in a form that is easy to understand and use.

Rationalization simplifies complex expressions and avoids dealing with irrational numbers in denominators.

Simplified Radical Form

The simplified radical form involves rewriting radicals in their most compact and easy-to-manage form. For instance, simplifying \(\sqrt{12} = 2 \sqrt{3}\) because 12 can be expressed as \(4 \cdot 3\), and \(\sqrt{4}\) is a perfect square. This step is crucial in finding the simplest form of a radical, as it involves breaking down the number into its factors to find the square roots.

Understanding the components inside a radical helps you simplify it correctly:

• Identify perfect square factors whenever possible.
• Simplify each part separately to get the simplest form.
• Recombine simplified parts to finalize the expression.

Remember, the goal is to have as few radicals and the smallest possible coefficients in the expression.

Algebraic Fractions

Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions, such as variables, constants, and radicals. Simplification of these fractions often involves both general fraction rules and the application of algebraic principles, including rationalization and simplification of radicals.

Working with algebraic fractions involves:

• Identifying factors in both the numerator and the denominator.
• Factoring expressions when possible to simplify them.
• Using rationalization effectively when dealing with radicals.

These techniques help you manipulate and simplify complex expressions, allowing you to express them in the simplest form possible. Simplifying fractions is a key skill in algebra, extensively used in solving equations and analyzing expressions.