Problem 18 - Algebra 2 and Trigonometry

Question

In \(3-38\) write each expression in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{24}+2 \sqrt{\frac{3}{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 3\sqrt{6} \).

Step 1: Simplify the Radicals

Begin by simplifying each radical separately. The square root of 24 can be simplified because 24 is equal to 4 times 6, and 4 is a perfect square. So, we have: \( \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \). The other term is \( 2\sqrt{\frac{3}{2}} \). The square root of \( \frac{3}{2} \) is simplified by finding the square root of the numerator and the denominator separately: \( \sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}} \).

Step 2: Rationalize the Denominator

To simplify \( \frac{\sqrt{3}}{\sqrt{2}} \), we need to rationalize the denominator. Multiply both the numerator and the denominator by \( \sqrt{2} \): \( \frac{\sqrt{3}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{3} \times \sqrt{2}}{2} = \frac{\sqrt{6}}{2} \). Thus, \( 2\sqrt{\frac{3}{2}} = 2 \times \frac{\sqrt{6}}{2} = \sqrt{6} \).

Step 3: Combine Like Terms

Now we have two terms: \( 2\sqrt{6} \) and \( \sqrt{6} \). These are like terms because they both involve the radical \( \sqrt{6} \). Combine them by adding their coefficients: \( 2\sqrt{6} + \sqrt{6} = 3\sqrt{6} \).

Step 4: Write the Expression in Simplest Form

After combining the like terms, the expression simplifies to \( 3\sqrt{6} \). This is the simplest form of the given expression.

Key Concepts

Rationalizing the Denominator

When you come across a radical term like \( \frac{\sqrt{3}}{\sqrt{2}} \), one useful technique is to rationalize the denominator. This means eliminating any square roots from the denominator. Why do we do this? It often makes numbers easier to work with and traditionally is the preferred way to write fractions.

To rationalize, multiply the fraction by a form of 1 that uses the radical in the denominator. Here, we use \( \frac{\sqrt{2}}{\sqrt{2}} \) because multiplying by 1 does not change the value of the expression:

• \( \frac{\sqrt{3}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{3} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} \)
• This simplifies to \( \frac{\sqrt{6}}{2} \) because \( \sqrt{2}\times \sqrt{2} = 2 \).

Now the denominator is a nice whole number, making calculations or further simplifications easier.

Combining Like Terms

Combining like terms is a crucial step in simplifying any algebraic expression. It involves adding or subtracting terms that have identical variables and powers.

For example, in the expression \( 2\sqrt{6} + \sqrt{6} \), both terms are like terms because they contain the radical \( \sqrt{6} \). Combining them requires only to add the coefficients in front of the identical radical:

• \( 2\sqrt{6} + \sqrt{6} = 3\sqrt{6} \).

Think of it like collecting apples. If you have 2 apples and someone gives you 1 more, you simply count all the apples together to end up with 3 apples. The same concept applies to the radicals.

Simplest Form of an Expression

The simplest form of an expression is when it is expressed as clearly and compactly as possible, eliminating any extraneous parts. The aim is to make it as easy to interpret and use as possible.

To achieve this, follow these steps:

• First, simplify each radical to reduce it to the smallest components. For example, \( \sqrt{24} \) becomes \( 2\sqrt{6} \) because \( 24 \) can be broken down to \( 4 \times 6 \), with \( \sqrt{4} = 2 \).
• Next, rationalize the denominator in any fraction that contains radicals. This avoids awkward division by irrational numbers.
• Finally, combine any like terms to achieve further simplification. In this example, \( 2\sqrt{6} + \sqrt{6} \) combines to \( 3\sqrt{6} \).

Through these steps, you achieve the most efficient and straightforward presentation of the expression.