Problem 221 - Elementary Algebra

Question

In the following exercises, simplify. $$ 21 \sqrt{19}-2 \sqrt{19} $$

Step-by-Step Solution

Answer

19 \sqrt{19}

Step 1: Identify Like Terms

Recognize that both terms have the same radical part \( \sqrt{19} \). This means they are like terms and can be combined.

Step 2: Combine the Coefficients

The expressions can be combined by adding the coefficients in front of the radicals. The coefficients are 21 and -2.

Step 3: Perform the Addition/Subtraction

Add the coefficients: \[ 21 - 2 = 19 \]. Thus, \[ 21 \sqrt{19} - 2 \sqrt{19} = 19 \sqrt{19} \].

Key Concepts

Like Terms

In algebra, 'like terms' are terms that contain the same variables raised to the same power. This means the coefficients (the numerical part of the terms) may differ, but the variable parts are identical. For instance, in the example provided, both terms include the same radical, \( \sqrt{19} \). Because of this, they are considered like terms. In practical terms, this allows us to combine them by simply adding or subtracting their coefficients. Think of it as combining apples with apples! You wouldn’t mix apples with oranges.

Radicals

A radical is an expression that includes a square root, cube root, or other roots. In our problem, the radical is \( \sqrt{19} \). Radicals can sometimes make simplifying expressions seem tricky. However, a key point to remember is that radicals must be the same to combine them as like terms. In simpler terms, if you have \( 21 \sqrt{19} \) and \( -2 \sqrt{19} \), they both revolve around the same radical part (\( \sqrt{19} \)). This means we can treat the question just like a basic algebra problem of combining like terms, only involving the coefficients.

Coefficients

Coefficients are the numerical part of terms in an expression. When simplifying expressions, coefficients can be added or subtracted when their corresponding variables or radicals are the same. For example, in the equation \( 21 \sqrt{19} - 2 \sqrt{19} \), 21 and -2 are the coefficients of \( \sqrt{19} \). By focusing on the numerical part, we treat it like a simple arithmetic problem: \( 21 - 2 = 19 \). So, the final expression becomes \( 19 \sqrt{19} \). Always remember: while coefficients can change through addition or subtraction, the radical or variable part remains consistent in these operations.