Problem 24 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The simplified form of the given expression is -\( \sqrt{6} \).

Step 1: Simplify each square root

First, break each number under the square root into a product of a perfect square and another number. The number 24 can be illustrated as \( 4 \times 6 \), 96 as \( 16 \times 6 \), and 6 stays as it is because it cannot be broken down further in this case.

Step 2: Apply the square root property

Apply the square root property that \( \sqrt{a*b} = \sqrt{a}*\sqrt{b} \). Thus \( \sqrt{24} \) becomes \( \sqrt{4}*\sqrt{6} = 2\sqrt{6} \), \( \sqrt{96} \) becomes \( \sqrt{16}*\sqrt{6} = 4\sqrt{6} \), and \( \sqrt{6} \) stays as it is.

Step 3: Substitution and Simplification

Replace the simplified roots into the original expression and simplify. The expression \( \sqrt{24}-\sqrt{96}+\sqrt{6} \) becomes \( 2\sqrt{6}-4\sqrt{6}+\sqrt{6} \).

Step 4: Combine Like terms

The terms \( 2\sqrt{6}, -4\sqrt{6} \), and \( \sqrt{6} \) can be combined because they are like terms (same radical part). The simplified form will be \( -\sqrt{6} \).

Key Concepts

Square Root Properties

When dealing with square roots, it's important to understand certain properties that make simplifying expressions easier. One crucial property is that the square root of a product can be separated into the product of square roots:

• If you have \( \sqrt{a \times b} \), it can be rewritten as \( \sqrt{a} \times \sqrt{b} \).

This property is particularly useful when breaking down numbers into factors, especially dividing them into perfect squares and other numbers. This helps in taking the square root of the perfect square separately to simplify the expression. For example, \( \sqrt{24} \) can be broken down as \( \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \). Understand and apply these properties for efficient simplification of square roots.

Perfect Squares

A perfect square is a number that is the square of an integer. Recognizing perfect squares is vital in simplifying square roots.

• For instance, numbers like 4, 9, 16, and 25 are perfect squares as they are \( 2^2, 3^2, 4^2, \) and \( 5^2 \), respectively.

In the solution process, expressing a number as a product of a perfect square simplifies the evaluation. For example, in \( \sqrt{96} \), recognizing 16 as a perfect square allows you to simplify it to \( \sqrt{16} \times \sqrt{6} = 4\sqrt{6} \). Breaking numbers into perfect squares aids in reducing the square root expressions more easily.

Like Terms

In algebra, like terms refer to terms that have the same variables and exponents. With radicals, 'like terms' have the same radicand.

• In the expression \( 2\sqrt{6} - 4\sqrt{6} + \sqrt{6} \), the terms are like terms because they all share the radical part \( \sqrt{6} \).

To simplify, add or subtract the coefficients (numbers in front of the radicals). Therefore, you calculate \( 2 - 4 + 1 \) which equals \(-1\). This means the expression simplifies to \(-\sqrt{6}\). By identifying and combining like terms, you can simplify radical expressions effectively.

Radicals in Algebra

Radicals often appear in algebra, especially under square roots. They can seem complex, but with practice, they are manageable.

• Radicals are expressions that include the root of a number, usually indicated by the square root sign \( \sqrt{} \).
• Handling radicals involves recognizing when numbers can be simplified and when they remain under the root.

For example, the expression \( \sqrt{24} - \sqrt{96} + \sqrt{6} \) uses the properties of radicals to simplify each part. It's useful to know that radicals can often be expressed in their simplest forms by breaking them into smaller parts, involving the square root properties and perfect squares. Simplifying radicals is an essential skill that helps students solve more complex algebraic problems with ease.