Problem 4 - Technical Mathematics with Calculus

Question

Addition and Subtraction of Radicals. Combine as indicated and simplify. $$\sqrt{128}-\sqrt{18}+\sqrt{32}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{128}-\sqrt{18}+\sqrt{32}\) is \(9\sqrt{2}\).

Step 1: Factor each radicand into its prime factors

Express 128, 18, and 32 as products of their prime factors. 128 = 2^7, 18 = 2 * 3^2, and 32 = 2^5.

Step 2: Rewrite each radical using perfect square factors

Break down each radicand into a product of squares and non-squares. \(\sqrt{128} = \sqrt{2^6 \cdot 2} = \sqrt{(2^3)^2 \cdot 2} = 2^3\sqrt{2} = 8\sqrt{2}\), \(\sqrt{18} = \sqrt{2 \cdot (3^2)} = 3\sqrt{2}\), \(\sqrt{32} = \sqrt{2^4 \cdot 2} = \sqrt{(2^2)^2 \cdot 2} = 2^2\sqrt{2} = 4\sqrt{2}\).

Step 3: Combine like radicals

Add or subtract the radicals with the same radicand. \(8\sqrt{2} - 3\sqrt{2} + 4\sqrt{2} = (8 - 3 + 4)\sqrt{2} = 9\sqrt{2}\).

Key Concepts

Addition and Subtraction of Radicals

When dealing with the addition and subtraction of radicals, it's important to understand that only 'like' radicals can be combined. Just like with traditional variables, radicals must have the same radicand—the number or expression under the radical sign—to be added or subtracted. It's similar to the idea that you can only add or subtract like terms such as apples with apples, not apples with oranges.

For example, \(\sqrt{8}\) and \(\sqrt{18}\) cannot be directly combined because their radicands are not the same. To simplify such expressions, you need to break down each radicand into its prime factors and simplify the radicals to the extent that their radicands match. Once you have like radicals, just add or subtract the coefficients in front of the radicals, and keep the radicand unchanged.

Prime Factorization

Prime factorization plays a critical role in simplifying radicals. By expressing the number under the radical sign as a product of its prime factors, you create an opportunity to identify and extract perfect squares that can be simplified outside the radical.

Every composite number can be expressed uniquely as a product of prime numbers, known as its prime factors. To simplify a square root, you should look for pairs of prime factors because \(\sqrt{p^2} = p\), where \(p\) is a prime number. In the case of \(\sqrt{128}\), you would write it as \(\sqrt{2^7}\), then further simplify to \(2^3\sqrt{2}\) or \(8\sqrt{2}\), as \(2^6\) is a perfect square. Understanding prime factorization is thus vital to breaking down and simplifying radical expressions.

Combining Like Terms

The concept of combining like terms extends to radical expressions as well. Once you have performed prime factorization and simplified the radicals, you can combine terms that have the same radicand, much like combining similar algebraic terms. For example, \(5x + 3x = 8x\) combines the like terms by adding their coefficients; this is analogous to how \(8\sqrt{2} - 3\sqrt{2} + 4\sqrt{2}\) can be combined to yield \(9\sqrt{2}\).

Remember the key steps are first to simplify the radicals as much as possible, and then proceed to combine only those terms with matching radicands. By treating the radical sign as a 'variable', and the number inside as its 'value', you can more easily visualize how like terms may be combined through addition or subtraction.