Problem 20 - Algebra 1

Question

Simplify the expression. $$11 \sqrt{3}-12 \sqrt{3}$$

Step-by-Step Solution

Answer

The simplified expression is \(-1 \sqrt{3}\).

Step 1: Identify Like Terms

Recognize the like terms in the expression which are \(11 \sqrt{3}\) and \(-12 \sqrt{3}\). These are called like terms as they have same square root, \( \sqrt{3} \).

Step 2: Combine Like Terms

Combine these like terms by performing the subtraction operation which gives \(-1 \sqrt{3}\). The further explains that negative sign specifies that \( \sqrt{3} \) is subtracted 1 times.

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical phrases into their most basic form. When it comes to radical expressions like \(11 \sqrt{3} - 12 \sqrt{3}\), our goal is to make it as simple as possible. Here, you are dealing with square roots, which are a common type of radical expressions.

To simplify, you first need to identify and focus on like terms. In this example, \(\sqrt{3}\) is the common radical factor in both parts of the expression. This means you're dealing with terms that have the same square root, and you only need to perform operations (such as addition or subtraction) on the numerical coefficients in front of the radicals. Simplification here reduces the expression without altering its overall value, making it more efficient to work with in further calculations.

Combining Like Terms

Combining like terms is a crucial step when working with radical expressions. It involves grouping together the terms that have identical variables or, in this case, radical parts. In the context of radicals, like terms have the same radical component.

For \(11 \sqrt{3} - 12 \sqrt{3}\), this means you're combining two terms that both include \(\sqrt{3}\). Because they share this common element, you can combine them mathematically. The process involves performing the arithmetic operation prescribed—here, subtraction—on the coefficients of these radicals: 11 and -12. The calculation \(11 - 12\) results in -1, which means the like terms combine to form \(-1 \sqrt{3}\). By doing this, you transform the expression into a more simplified and manageable form.

Subtraction of Radicals

When subtracting radicals, ensure that the radicals have the same root value. If they do, as in \(11 \sqrt{3} - 12 \sqrt{3}\), you follow the standard rules of arithmetic. Unlike whole numbers or other algebraic expressions, radicals require you to pay attention to their radicands (the number under the root).

The subtraction process is straightforward:

• Identify radicals with the same radicand.
• Subtract the coefficients while keeping the radical part unchanged.

Once you know that the radicals have the same \(\sqrt{3}\), subtract their coefficients: \(11 - 12 = -1\). The result \(-1 \sqrt{3}\) shows that you have essentially subtracted these like terms successfully. Maintaining accuracy in each step significantly streamlines any further work you do with these expressions in broader mathematical problems.