Problem 65 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. All variables represent positive real numbers. $$ 4 \sqrt{2 x}+6 \sqrt{2 x} $$

Step-by-Step Solution

Answer

The simplified expression is \\(10 \sqrt{2x}\\).

Step 1: Identify Like Radicals

Examine the expression to determine if the radicals have the same radicand and index. Both terms are radicals with \(2x\) under the square root, making them like radicals.

Step 2: Add the Coefficients

Since the radicals are the same, we can add their coefficients together. Therefore, add 4 and 6 to get the combined coefficient: \(4 + 6 = 10\).

Step 3: Write the Simplified Expression

Combine the like radicals by placing the sum of the coefficients in front of the common radical. The simplified expression becomes \(10 \sqrt{2x}\).

Key Concepts

Combining Like Terms

When you're simplifying expressions, especially those involving radicals, it's important to understand the concept of combining like terms. In the case of radicals, like terms are those that have the same radicand (the number or expression inside the radical root) and the same index (the degree of the root). For example, in the expression \(4 \sqrt{2x} + 6 \sqrt{2x}\), both terms are like radicals because they share the same radicand, \(2x\), and have the same index, which is 2 since it's a square root.

Once you identify like terms, you can combine them by adding or subtracting their coefficients (the numbers in front of the radicals). This process is similar to combining like terms in general algebra, where variables must match exactly before combining. Combining like radicals follows the same rules: you perform arithmetic operations only on the coefficients while the radical part remains unchanged. So, \(4 \sqrt{2x} + 6 \sqrt{2x}\) simplifies to \((4 + 6) \sqrt{2x} = 10 \sqrt{2x}\).

If the radicands or indices are different, the radicals cannot be combined in this simple manner. Understanding this differentiation is crucial to simplifying expressions correctly.

Radicals

Radicals are expressions that include a root symbol. The most common radical is the square root, indicated by \(\sqrt{}\), but there are also cube roots, fourth roots, and so on. Each radical has two main components: the radicand, the number or expression inside the radical; and the index, which tells you the degree of the root (e.g., square, cube).

Working with radicals requires an understanding of properties like:

• The product rule: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
• The quotient rule: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)
• Simplifying a square root: Find the prime factors of the radicand to simplify \(\sqrt{a}\)

Applying these rules helps simplify complex radical expressions and identify like radicals which you can then combine or simplify further. For expressions like \(\sqrt{2x}\), simplifying further might not be possible unless \(x\) is known or factorable.

Square Roots

Square roots are special kinds of radicals where the index is 2. This means you're looking for a number which, when multiplied by itself, equals the radicand. The square root of a number \(k\) is written as \(\sqrt{k}\).

For practical use:

• If \(k\) is a perfect square (like 4, 9, 16), its square root is an integer (like 2, 3, 4).
• For non-perfect squares, the square root is an irrational number that can often be approximated.
• A common property of square roots is \((\sqrt{a})^2 = a\), which provides a way to reverse the operation of taking a square root.

Understanding how to work with square roots is fundamental to dealing with radical expressions. Functions involving square roots can represent geometric problems, such as finding the side length of a square given its area. Simplifying expressions with square roots often requires recognizing when numbers can be factored into perfect squares, allowing further simplification.