Problem 19 - College Algebra

Question

Simplify each expression. $$14 \sqrt{6}-6 \sqrt{24}$$

Step-by-Step Solution

Answer

The simplified expression is \(2 \sqrt{6}\).

Step 1: Simplify the Square Roots

Begin by simplifying the square roots in the expression. For \(\sqrt{24}\), we factor it down to its prime factors: \(24 = 4 \times 6 = 2^2 \times 6\). We can take \(\sqrt{4}\) out of the square root because \(\sqrt{4} = 2\). Thus, \(\sqrt{24} = 2 \sqrt{6}\).

Step 2: Substitute Simplified Square Root in Expression

Substitute the simplified square root \(\sqrt{24} = 2 \sqrt{6}\) back into the original expression. Thus, the expression \(14 \sqrt{6} - 6 \sqrt{24}\) becomes \(14 \sqrt{6} - 6 \times 2 \sqrt{6}\).

Step 3: Simplify the Expression by Combining Like Terms

In the expression \(14 \sqrt{6} - 6 \times 2 \sqrt{6}\), multiply the terms: \(6 \times 2 \sqrt{6} = 12 \sqrt{6}\). Then the expression becomes \(14 \sqrt{6} - 12 \sqrt{6}\).

Step 4: Complete the Simplification by Subtracting Like Terms

Subtract the like terms: \(14 \sqrt{6} - 12 \sqrt{6} = 2 \sqrt{6}\). Thus, the simplified form of the expression is \(2 \sqrt{6}\).

Key Concepts

Understanding Square Roots

Square roots are a fundamental concept in mathematics. They help us find a number which, when multiplied by itself, yields the original number underneath the square root symbol. For example, the square root of 9, denoted as \(\sqrt{9}\), is 3, because 3 times 3 equals 9. Knowing how to simplify square roots is essential especially when dealing with expressions that include terms under a radical.To simplify a square root like \(\sqrt{24}\), break it down into its prime factors. Prime factorization helps identify perfect square factors. For \(24\), the prime factorization is \(2^3 \times 3\). We can then extract any perfect squares, such as \(2^2\), from under the square root. Since \(2^2 = 4\) and \(\sqrt{4} = 2\), \(\sqrt{24}\) simplifies to \(2\sqrt{6}\) (\(\sqrt{2^2 \times 6} = 2\sqrt{6}\)). This process is valuable in simplifying expressions that contain square roots.

Mastering Combining Like Terms

Combining like terms is key in simplifying algebraic expressions. Like terms have identical variable parts or radical parts, even if their coefficients are different. If you look at \(14 \sqrt{6} - 12 \sqrt{6}\), both terms are like terms because they share the common square root factor \(\sqrt{6}\).When combining like terms, you simply add or subtract their coefficients. In the case of \(14 \sqrt{6} - 12 \sqrt{6}\), you treat \(\sqrt{6}\) as a common factor and subtract the coefficients 14 and 12, which leaves you with \(2 \sqrt{6}\). Think of this process like collecting similar items, where you group terms that "belong" together to make calculations easier.

The Role of Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime factors. This is a helpful technique when simplifying square roots. Understanding that numbers can be broken into prime factors lets you easily identify perfect squares, which can simplify radical expressions.For instance, to simplify \(\sqrt{24}\), you analyze 24 by breaking it into prime numbers: \(2 \times 2 \times 2 \times 3\) or \(2^3 \times 3\). The prime factorization exposes \(2^2\) a perfect square, which can be taken outside the square root as 2 since \(\sqrt{2^2} = 2\). This leaves you with \(2\sqrt{6}\) (as \(2\) is taken out and \(6\) remains under the square root).This technique of using prime factorization supports efficient simplification of radical expressions, making calculations much more manageable.