Problem 166 - Elementary Algebra

Question

In the following exercises, simplify. $$ 5 \sqrt{7}-8 \sqrt{7}+6 \sqrt{3} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-3 \sqrt{7} + 6 \sqrt{3}\).

Step 1: Identify Like Terms

The expression contains terms with square roots. Identify terms with the same square root. Here, \(5 \sqrt{7}\), \(-8 \sqrt{7}\) are like terms. \(6 \sqrt{3}\) is a different term.

Step 2: Combine Like Terms

Combine the coefficients of the like terms \(5 \sqrt{7}\) and \(-8 \sqrt{7}\). This gives: 5 - 8 = -3. Therefore, \(5 \sqrt{7} - 8 \sqrt{7} = -3 \sqrt{7}\).

Step 3: Write the Final Simplified Expression

Combine the simplified term \(-3 \sqrt{7}\) with the remaining term. Thus, the final expression is \(-3 \sqrt{7} + 6 \sqrt{3}\).

Key Concepts

like terms

When simplifying radical expressions, it is important to identify like terms. Like terms in radical expressions have the same radicand (the number under the square root). In the given exercise, we have three terms: \(5 \sqrt{7}\), \(-8 \sqrt{7}\), and \(6 \sqrt{3}\). Here, \(\sqrt{7}\) is the common radicand for the first two terms, making them like terms. The third term \(6 \sqrt{3}\) has a different radicand (\(\sqrt{3}\)), therefore, it is not a like term with the first two. This step sets the stage for combining like terms, streamlining our simplification process.

combining coefficients

After identifying like terms, the next step is combining their coefficients. Coefficients are the numerical factors in front of the radicals. In the expression \(5 \sqrt{7}-8 \sqrt{7}+6 \sqrt{3}\), the coefficients to be combined are 5 and -8 for the like terms \(5 \sqrt{7}\) and \(-8 \sqrt{7}\). Subtracting 8 from 5 (\(5 - 8\)), results in -3. Therefore, \(5 \sqrt{7} - 8 \sqrt{7}\) simplifies to \(-3 \sqrt{7}\). The term \(6 \sqrt{3}\) remains unchanged as it does not have any like terms to combine with.

radicals

Radicals are expressions that include a root, such as a square root or a cube root. In this exercise, all the terms contain square roots, which are a type of radical. Radicals themselves cannot be combined unless they have the same radicand. This principal guides us to separately handle terms with different radicands. After simplifying the like terms \(5 \sqrt{7}\) and \(-8 \sqrt{7}\) to \(-3 \sqrt{7}\), we combine this result with the remaining term \(6 \sqrt{3}\). Thus, our final simplified expression is \(-3 \sqrt{7} + 6 \sqrt{3}\). Understanding radicals and their properties is key to mastering the simplification of such expressions.