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Chapter 8: Problem 21

Simplify. $$\sqrt{18}$$

Short Answer

Expert verified
3\sqrt{2}

Step by step solution

01

- Factorize the number under the square root

Identify the factors of 18. The factors are 18 = 2 * 3 * 3 = 2 * 3^2.
02

- Separate the factors in the square root

Rewrite the square root separating the perfect square: \ \ \ \(\sqrt{18} = \sqrt{2 * 3^2}\).
03

- Simplify the square root

Take the square root of the perfect square part out: \ \ \ \(\sqrt{2 * 3^2} = 3\sqrt{2}\).

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Key Concepts

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

Factorization
Factorization is a key technique used in simplifying square roots. It involves breaking down a number into its prime factors. For instance, to simplify \(\text{鈭�18}\), we need to identify its prime factors. The factors of 18 are 2 and 3, specifically: \(\text{18 = 2 * 3 * 3 = 2 * 3}^2\). This step is crucial because it reveals any perfect square factors within the number. Always look for pairs of the same number, as they signify perfect squares. Remember, factorization helps us separate and eventually simplify the square root.
Perfect Squares
Perfect squares play a significant role in simplifying square roots. A perfect square is a number that can be expressed as the product of an integer with itself. For example, \(\text{3}^2 = 9\) and \(\text{4}^2 = 16\) are perfect squares. In the context of \(\text{鈭�18}\), we find that 18 contains \(\text{3}^2\) as part of its factorization: \(\text{鈭�(2 * 3}^2)\). Identifying the perfect square helps in transforming the expression into a simpler form. Always extract the perfect square from under the radical sign. This makes the square root more manageable and easy to work with.
Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, etc. Simplifying these expressions means reducing them to their simplest form. As seen in \(\text{鈭�18}\), once we have factored it and identified the perfect square, we can rewrite it as: \(\text{鈭�(2 * 3}^2)\). We then take the square root of the perfect square part (3) out of the radical, giving us \(\text{3鈭�2}\). Simplifying radical expressions makes them easier to understand and work with in mathematical problems. Always follow these steps: factorize, identify perfect squares, and simplify the expression.

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Most popular questions from this chapter

Simplify. $$x^{\frac{2}{9}} x^{\frac{3}{5}} x^{\frac{1}{6}}$$

(a) solve. Choose a window that shows the point of intersection. Sketch the graph; describe the window. (b) identify any solutions. $$\sqrt{2 x+18}=4$$

Evaluate. $$\sqrt[4]{-81}$$

The completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(\sqrt{3 x}=-15\)

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