Problem 29 - Algebra 1

Question

Simplify the expression. $$\sqrt{24}-\sqrt{96}+\sqrt{6}$$

Step-by-Step Solution

Answer

\(-\sqrt{6}\)

Step 1: Simplify the square roots

Start by simplifying each square root expression. The square root of 24 can be broken down as \(\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}\). Similarly, \(\sqrt{96} = \sqrt{16 \cdot 6} = \sqrt{16} \cdot \sqrt{6} = 4\sqrt{6}\). The square root of 6 remains as is. So, the given expression becomes \(2\sqrt{6} - 4\sqrt{6} + \sqrt{6}\).

Step 2: Perform the operations

Now, perform the addition and subtraction operations on the simplified expressions. The terms \(2\sqrt{6}\), \(-4\sqrt{6}\), and \(\sqrt{6}\) can be combined because they are like terms. Their coefficients are added together: \(2\sqrt{6} - 4\sqrt{6} + \sqrt{6} = -\sqrt{6}\).

Key Concepts

Radical Expressions

When we talk about radical expressions, we're referring to mathematical phrases that include a radical symbol, \( \sqrt{} \), which denotes the square root of a number. These expressions represent the number that, when multiplied by itself, will give you the original number under the radical. For instance, \( \sqrt{9} \) equals 3 because 3 multiplied by itself is 9.

Simplifying radical expressions involves breaking them down into simpler components, if possible. As seen in our exercise, \( \sqrt{24} \) was simplified by finding the largest square number that divides 24, which is 4, and expressing 24 as \( 4 \times 6 \). The square root of a product, like \( \sqrt{4 \times 6} \) can be expressed as \( \sqrt{4} \cdot \sqrt{6} \) or \( 2\sqrt{6} \) because \( \sqrt{4} \) is 2. This step makes it easier to work with the expressions when combining them later on.

Like Terms

In algebra, like terms are terms that have the exact same variable parts raised to the same power. The only difference between them is their coefficients. For example, \( 3x \) and \( 5x \) are like terms because they both contain the variable \( x \) raised to the same power, even though they have different coefficients, 3 and 5 respectively. This concept is crucial when simplifying expressions because like terms can be combined through addition or subtraction.

In the context of radical expressions, like terms have the same radical part. So, \( 2\sqrt{6} \) and \( -4\sqrt{6} \) are like terms because they are both the square root of 6, even though they have different coefficients of 2 and -4. Recognizing like terms allows us to combine them into a single term to simplify the expression.

Simplifying Radicals

The process of simplifying radicals aims to find the simplest radical term possible. This involves identifying and removing perfect square factors from under the radical sign. As seen in the exercise, \(\sqrt{96}\) can be simplified by recognizing that 96 is divisible by the perfect square 16. We can rewrite the expression as \(\sqrt{16} \cdot \sqrt{6}\) or \(4\sqrt{6}\) because \(\sqrt{16}\) equals 4.

When simplifying radicals, look for the largest square number that can be factored out of the radicand—the number under the radical. Breaking down the radical into its simplest form not only makes it easier to understand but also sets the stage for combining like terms.

Combining Like Terms

Once radicals are simplified, the next step is combining like terms. This can only be done with terms that have the same radical components. In our original exercise, each term contained a \(\sqrt{6}\), making them like terms. When combining like terms with radicals, just add or subtract their coefficients.

In the exercise, \(2\sqrt{6}\) and \(4\sqrt{6}\) were different only in their coefficients. They were combined by adding their coefficients: \(2 - 4 + 1\), which resulted in \( -1\sqrt{6}\) or \( -\sqrt{6}\). This concept is vital for simplifying and solving equations where radicals are present and is a fundamental skill in algebra.