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Chapter 7: Problem 21

Add or subtract to simplify each radical expression. Assume that all variables represent positive real numbers. \(\sqrt{72 x}-\sqrt{8 x}\)

Short Answer

Expert verified
4\sqrt{2x}

Step by step solution

01

Simplify the Radicals

To simplify \(\sqrt{72x} - \sqrt{8x}\), each radical should be simplified separately. Start with \(\sqrt{72x}\): \(\sqrt{72x} = \sqrt{36 \cdot 2x} = \sqrt{36} \cdot \sqrt{2x} = 6\sqrt{2x}\). Hence, \(\sqrt{72x} = 6\sqrt{2x}\). Now simplify \(\sqrt{8x} = \sqrt{4 \cdot 2x} = \sqrt{4} \cdot \sqrt{2x} = 2\sqrt{2x}\). Hence, \(\sqrt{8x} = 2\sqrt{2x}\).
02

Express the Radicals in Their Simplified Form

Replace \(\sqrt{72x}\) and \(\sqrt{8x}\) in the original expression with their simplified forms. This gives you: \(\sqrt{72x} - \sqrt{8x} = 6\sqrt{2x} - 2\sqrt{2x}\).
03

Combine Like Terms

Combine the like terms \(6\sqrt{2x}\) and \(2\sqrt{2x}\). Subtract them: \((6 - 2)\sqrt{2x} = 4\sqrt{2x}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Radicals
Radicals are expressions that include a root symbol, often a square root (\sqrt{}). Simplifying radicals involves reducing them to their simplest form. To do this, identify factors that are perfect squares. For example, \(\sqrt{72x}\) can be simplified to \(\sqrt{36 \cdot 2x}\). Since \(\sqrt{36} = 6\), we can rewrite \(\sqrt{72x}\) as \(6\sqrt{2x}\). This process helps in managing and solving equations more easily. It鈥檚 crucial to become comfortable breaking down numbers and variables to their simplest radical form.
Combining Like Terms
Like terms are terms that have identical variable parts. In the context of radicals, this means the terms must have the same expression under the radical sign. For instance, once \(\sqrt{72x}\) and \(\sqrt{8x}\) are simplified to \(6\sqrt{2x}\) and \(2\sqrt{2x}\) respectively, they can be combined because they share the \(\sqrt{2x}\) term. Combining them involves straightforward addition or subtraction of their coefficients. In the example provided, you subtract the coefficients: \((6 - 2)\sqrt{2x} = 4\sqrt{2x}\). This step is essential for making expressions more manageable and for solving equations effectively.
The Simplification Process
The simplification process involves several steps:
  • Simplify each radical expression individually by identifying and extracting perfect square factors.
  • Simplify resulting radicals to the form \(a\sqrt{b}\) where \(a\) is a coefficient and \(b\) remains under the radical.
  • Express simplified radicals in the given expression.
  • Combine like terms by adding or subtracting their coefficients.
Following these steps ensures a clear path to simplifying complex radical expressions. For example, in \(\sqrt{72x} - \sqrt{8x}\), simplifying each term individually results in \(6\sqrt{2x}\) and \(2\sqrt{2x}\). By recognizing these as like terms, you can then combine them to form \(4\sqrt{2x}\). This methodical approach helps clarify each part of the problem and leads to a simplified solution.

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