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In algebra, the concept of like terms is crucial when simplifying expressions. Like terms are terms within an expression that have the same variable component and/or the same root, if dealing with radicals. For example, in the exercise \( \sqrt{3} - 5 \sqrt{7} - 12 \sqrt{3} \), the terms \( \sqrt{3} \) and \( -12 \sqrt{3} \) are considered like terms because they both contain the same radical, \( \sqrt{3} \). However, \( -5 \sqrt{7} \) is not a like term with the others because it is associated with a different radical, \( \sqrt{7} \).

Understanding like terms helps you to see which parts of an expression can be directly combined. This process simplifies expressions and makes it easier to solve or further manipulate algebraic equations. Always look for identical roots or variables with the same powers. If they match, they are like terms!

Once you've identified the like terms in an expression, the next step is to combine them. This is done through basic addition or subtraction. In the context of the exercise \( \sqrt{3} - 5 \sqrt{7} - 12 \sqrt{3} \), we combine \( \sqrt{3} \) and \( -12 \sqrt{3} \) because they are like terms.

Here's a detailed way to combine the like terms:

• Place the coefficients in front of the like terms. In this case, the coefficients are 1 for \( \sqrt{3} \) and -12 for \( -12 \sqrt{3} \).
• The operation becomes \( 1 - 12 \) which equals \( -11 \).
• Therefore, the expression becomes \( -11 \sqrt{3} \).

By combining like terms, you simplify the expression, reducing the number of terms and making it clearer, as seen in the resulting \( -11 \sqrt{3} - 5 \sqrt{7} \).

Radical expressions involve roots, such as square roots, cube roots, etc., and can often appear complex. However, the process of simplifying radical expressions usually involves identifying like terms and then combining them, as shown in our exercise.

Let's explore some key points about radical expressions:

• Radicals are the symbols we use for roots, for example, \( \sqrt{} \) for square roots.
• To simplify, first check for any like terms with the same radicand (the number or expression under the radical sign).
• Combine these like terms by adding or subtracting their coefficients.
• Sometimes, the expression may include radicals that cannot be simplified further, like our \( -5 \sqrt{7} \) which stands alone.

Working with these expressions means being comfortable with both numerical manipulation and understanding radical rules. Over time, this will make tackling more complex equations an easier task.