Problem 224 - Elementary Algebra

Question

In the following exercises, simplify. $$ 11 \sqrt{11}-10 \sqrt{11} $$

Step-by-Step Solution

Answer

\(\sqrt{11}\)

Step 1 - Identify Like Terms

Notice that both terms involve the square root of 11: \(11 \sqrt{11}\) and \(10 \sqrt{11}\). These can be treated as like terms in a similar way to how you would combine coefficients of the same variable.

Step 2 - Subtract the Coefficients

Subtract the coefficients of \(\sqrt{11}\). The coefficients are 11 and 10: \(11 - 10 = 1\)

Step 3 - Combine the Result with the Common Factor

Now combine the result with \(\sqrt{11}\): \(1 \cdot \sqrt{11} = \sqrt{11}\)

Key Concepts

Like Terms

When simplifying expressions with radicals, it's crucial to identify like terms. Like terms are terms that have the same variable factors.
In the given exercise, both terms involve the square root of 11:

• \( 11 \sqrt{11} \)
• \( 10 \sqrt{11} \)

Just like how you can combine terms such as \(2x\) and \(3x\) by adding or subtracting the coefficients (because they both contain x), you can combine terms like \( 11 \sqrt{11} \) and \( 10 \sqrt{11} \) as well. Here, \( \sqrt{11} \) acts like the variable part.

Only terms with the same radical part (in this case, \( \sqrt{11} \)) can be treated as like terms. Identifying like terms effectively simplifies the process of combining them.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5 because \( 5 \times 5 = 25 \).

Radicals often appear in simplification problems. The square roots allow you to break down numbers into simpler components and work with them in a more manageable way.
In our exercise, we're dealing with \( \sqrt{11} \), which means we're working with the principal square root of 11. We combine the terms involving \( \sqrt{11} \) just like how we would combine variables.

Understanding square roots helps you simplify and solve problems more effectively. It is an essential skill in algebra and beyond.

Coefficients

Coefficients are the numerical parts of terms that are multiplied by variables or radicals. In the term \( 11 \sqrt{11} \), the coefficient is 11.

• For \( 11 \sqrt{11} \), the coefficient is 11
• For \( 10 \sqrt{11} \), the coefficient is 10

When simplifying expressions, adding or subtracting like terms involves working with the coefficients. In our exercise, the coefficients are 11 and 10. Subtracting these gives you \( 11 - 10 = 1 \).
This result then needs to be re-multiplied by the common radical term (\( \sqrt{11} \)), giving you \( 1 \times \sqrt{11} = \sqrt{11} \).

Identifying and performing operations with coefficients correctly is a key part of simplifying algebraic expressions involving radicals.
Remember, the coefficient tells how many times you are taking the variable or the radical part of the term.