Problem 72 Simplify. Assume that all variab... [FREE SOLUTION]

Chapter 10: Problem 72

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{18 m^{2}}\)

Short Answer

Expert verified

\(3m \sqrt{2}\)

Step by step solution

- Factor the Radicand

Break down the expression under the square root. Identify the factors of 18 and rewrite the expression in terms of its prime factors: \(\sqrt{18 m^{2}} = \sqrt{2 \times 3^{2} \times m^{2}} \)

- Separate the Factors

Rewrite the square root of the product as the product of the square roots: \(\sqrt{2 \times 3^{2} \times m^{2}} = \sqrt{2} \times \sqrt{3^{2}} \times \sqrt{m^{2}} \)

- Simplify Each Square Root

Simplify the square roots that can be simplified: \(\sqrt{3^{2}} = 3\) and \(\sqrt{m^{2}} = m\). This yields: \(\sqrt{2} \times 3 \times m \)

- Combine the Simplified Terms

Multiply the simplified terms together: \(3m \sqrt{2}\)

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Key Concepts

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

prime factorization

Prime factorization is the process of expressing a number as the product of its prime factors. For example, the number 18 can be broken down into its prime factors: \(2 \times 3^{2}\). By understanding prime factorization, you can simplify complex algebraic expressions, especially when dealing with square roots.

square root properties

The square root of a number is a value that, when multiplied by itself, gives the original number. Important properties to remember include:
- The product rule: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
- Simplifying a perfect square: \(\text{If} \sqrt{x^2} = |x|\)
These properties help in breaking down and simplifying roots in algebra

simplification steps

To simplify square roots, follow these steps:
1. Prime factorize the number under the root.
2. Use the product rule to separate the factors.
3. Simplify the roots of perfect squares.
4. Combine and simplify the remaining terms.
Using these steps ensures that you simplify square roots accurately and confidently

algebraic expressions

Algebraic expressions often involve variables and constants under a square root. To simplify, treat variables like numbers:
1. Factorize both numbers and variables.
2. Apply square root rules uniformly.
Always assume variables represent positive numbers to avoid complications with absolute values
This approach helps in dealing with more complex algebraic equations and improves your problem-solving skills

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91影视

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{18 m^{2}}\)

Short Answer

Step by step solution

- Factor the Radicand

- Separate the Factors

- Simplify Each Square Root

- Combine the Simplified Terms

Key Concepts

prime factorization

square root properties

simplification steps

algebraic expressions

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