Problem 43 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. $$ \frac{3}{\sqrt{48}} $$

Step-by-Step Solution

Answer

The simplified format of the given radical expression is \( \frac{\sqrt{3}}{4} \)

Step 1: Simplify the denominator

Factor 48 into a perfect square and another integer. 48 can be expressed as \(16*3\), and when taking the square root of 16, we consider \( \sqrt{16*3} \) = \( \sqrt{16} * \sqrt{3} \)

Step 2: Evaluate the square root of 16

Simplify \( \sqrt{16} \), which equals 4. This turns the expression into \(3/(4*\sqrt{3})\)

Step 3: Simplify the expression

The resulting expression is \( \frac{3}{4\sqrt{3}} \). However, we usually don't leave a square root in the denominator, so we multiply both the numerator and denominator by \( \sqrt{3} \), obtaining \( \frac{3\sqrt{3}}{12} \)

Step 4: Simplify the fraction

To simplify the fraction \( \frac{3\sqrt{3}}{12} \), divide the numerator and the denominator by 3, leading to the solution of \( \frac{\sqrt{3}}{4} \)

Key Concepts

Radical Expressions

Radical expressions involve roots, typically square roots, but can include other roots like cube roots. Understanding these expressions is important because they frequently appear in algebra and other areas of mathematics.

• At the core of radical expressions is the radical symbol \( \sqrt{} \), which denotes the square root or other roots depending on the index.
• The expression inside the radical symbol is called the radicand.

A radical expression can be simplified by looking for factors of the radicand that are perfect squares (for square roots). This simplification process makes the expression easier to work with.
For example, in simplifying \( \frac{3}{\sqrt{48}} \), we factor 48 into 16 and 3, where 16 is a perfect square. Finding such factors is a great way to start simplifying radical expressions.

Perfect Squares

Perfect squares are numbers like 1, 4, 9, 16, 25, etc., which are the result of squaring an integer.

• They are crucial in simplifying radical expressions because they allow us to radically reduce parts of the expression.
• Recognizing perfect squares quickly improves efficiency when dealing with radicals.

When we identify a perfect square factor within a radicand, we can take its square root, effectively moving it outside of the radical. For instance, in the expression \( \sqrt{48} \), we recognize 16 as \( 4^2 \). This lets us take out \( \sqrt{16} \), which is 4, simplifying \( \sqrt{48} \) to \( 4\sqrt{3} \). Finding and using perfect square factors is an essential technique in simplifying expressions.

Rationalizing the Denominator

Rationalizing the denominator is a method used to eliminate radicals from the denominator of a fraction. This is a convention followed for clarity and simplicity in algebraic expressions.

To rationalize a denominator, multiply both the numerator and the denominator by a radical that will make the denominator a perfect square under the radical.

• As shown in the expression \( \frac{3}{4\sqrt{3}} \), we multiply by \( \sqrt{3}/\sqrt{3} \).
• This process turns the expression into \( \frac{3\sqrt{3}}{12} \), with the denominator becoming 12 as a result of \( \sqrt{3} \times \sqrt{3} = 3 \).

This step ensures the denominator is an integer. Simplifying any resulting fractions is the last step, as seen in the provided example where \( \frac{3\sqrt{3}}{12} \) simplifies to \( \frac{\sqrt{3}}{4} \) by reducing the fraction.