Problem 163 - Elementary Algebra

Question

In the following exercises, simplify. $$ 3 \sqrt{11}+2 \sqrt{11}-8 \sqrt{11} $$

Step-by-Step Solution

Answer

-3 \( \sqrt{11} \)

Step 1 - Identify Like Terms

All the terms in the expression involve \( \sqrt{11} \). Since they all have the same radical component, they are like terms and can be combined.

Step 2 - Add and Subtract Coefficients

Combine the coefficients of \( \sqrt{11} \). The expression is \( 3 \sqrt{11} + 2 \sqrt{11} - 8 \sqrt{11} \). This means adding \(3, 2 \) and then subtracting 8: \( 3 + 2 - 8 \).

Step 3 - Simplify the Coefficients

Calculate \( 3 + 2 - 8 \). This equals \( -3 \). Therefore, the expression is now \( -3 \sqrt{11} \).

Key Concepts

Like Terms in Algebra

Understanding like terms is essential when working with algebraic expressions. Like terms refer to terms that have the same variables raised to the same power. In the context of radicals, like terms mean the radicals themselves must be the same for the terms to be combined.

For example, in the expression \(3 \, \text{sqrt}(11) + 2 \, \text{sqrt}(11) - 8 \, \text{sqrt}(11)\), each term involves \( \text{sqrt}(11)\). This makes all of them like terms because they share the same radical component.
By recognizing like terms, you can simplify your expressions more effectively.

Combining Coefficients

Combining coefficients is the next step after identifying like terms. The coefficient is the numerical part that multiplies the variable or radical. In our example, the coefficients are 3, 2, and -8 for the \( \text{sqrt}(11)\) terms.

• 3 is the coefficient of the first term
• 2 is the coefficient of the second term
• -8 is the coefficient of the third term

To combine these coefficients, simply add or subtract them as indicated:
\ 3 + 2 - 8.\br>Simplifying these numbers gives \(3 + 2 = 5 \), and \(5 - 8 = -3\).
So, our expression now looks like \(-3 \, \text{sqrt}(11)\). Combining coefficients helps to consolidate terms and make the expression more manageable.

Radical Simplification

Radical simplification aims to reduce expressions involving radicals to their simplest form. In the given example, since all terms already involved the same radical, \( \text{sqrt}(11)\), the main task was combining the coefficients.
However, if we had terms involving different radicals, simplifying the radicals themselves might be necessary first. But in our specific problem, that was not required because all radicals were already identical.
After combining like terms and their coefficients, we simplified our expression to \( -3 \, \text{sqrt}(11)\). This is the simplest form of the given expression.
By following these steps: identifying like terms, combining coefficients, and simplifying radicals, you can tackle similar problems effectively.