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Chapter 9: Problem 95

Simplify. $$ \sqrt{192 x}-\sqrt{75 x} $$

Short Answer

Expert verified
3 \sqrt{3x}

Step by step solution

01

Factor the radicands

Factorize the numbers inside the square roots. For 192, use the prime factorization approach such that 192 = 64 脳 3 = 8^2 脳 3. For 75, use 75 = 25 脳 3 = 5^2 脳 3.
02

Simplify each square root

Apply the property \( \sqrt{a^2 b} = a \sqrt{b} \). Thus, \( \sqrt{192 x} = \sqrt{64 脳 3x} = 8 \sqrt{3x} \) and \( \sqrt{75 x} = \sqrt{25 脳 3x} = 5 \sqrt{3x} \).
03

Subtract the simplified terms

Combine the simplified terms: \( 8 \sqrt{3x} - 5 \sqrt{3x} \). This results in \( (8-5) \sqrt{3x} = 3 \sqrt{3x} \).

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Key Concepts

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

Radicands
A radicand is the number or expression inside the radical symbol (square root). For example, in the expression \(\root{\root{192x}}\), the radicand is \(192x\). When simplifying radicands, it helps to break them down into their factors. This process often involves prime factorization.

Radicands can sometimes contain variables, like in the term \(192x\). The variable x is part of the radicand and must be carried through the simplification process.

When handling radicands, remember that the goal is to simplify them into a form that makes performing operations, like subtraction or addition, straightforward.
Prime Factorization
Prime factorization is the process of breaking down a number into its prime number components. For example, the number 192 can be expressed as \(8^2 \times 3\) after factoring. Factorizing helps simplify square roots.

Here is a quick guide to prime factorizing 192:
  • 192 = 2 脳 96
  • 96 = 2 脳 48
  • 48 = 2 脳 24
  • 24 = 2 脳 12
  • 12 = 2 脳 6
  • 6 = 2 脳 3
When all factors are written out: \(192 = 2^6 \times 3\). Similarly, factorize 75 to get \(5^2 \times 3\).
This process ensures that we can easily apply square roots. So, using the earlier expressions: \(\sqrt{192x} \text{ becomes } 8\sqrt{3x} \).
Combining Like Terms
After simplifying the square roots, terms with the same base radicands must be combined. For example, \(8\sqrt{3x} - 5\sqrt{3x} \) are 'like' terms because both have the radicand \(3x\).

Combining them involves simple addition or subtraction of the coefficients (numbers in front of the radicals). In our example, \(8 - 5 = 3\), so combining these terms yields \(3\sqrt{3x}\).

Always remember to keep the radicand the same when performing such operations, as changing the radicand changes the entire term.

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