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Chapter 10: Problem 48

Simplify \(\sqrt{18}\)

Short Answer

Expert verified
The simplified form of \sqrt{18} is 3 \sqrt{2}.

Step by step solution

01

Factorize the number inside the square root

18 can be factorized into its prime factors. 18 = 2 脳 3 脳 3.
02

Identify perfect squares

Recognize that 18 contains the square of 3 since 3 脳 3 = 9 and 9 is a perfect square.
03

Separate the perfect square

Write the expression under the square root as the product of the perfect square and another factor: \[\begin{equation} \sqrt{18} = \sqrt{9 \times 2} \end{equation}\]
04

Simplify the square root of the perfect square

Take the square root of 9, which is 3. The expression now becomes: \[\begin{equation} \sqrt{18} = 3 \times \sqrt{2} \end{equation}\]
05

Write the simplified expression

Combine the results to write the final simplified form: \[\begin{equation} \sqrt{18} = 3 \sqrt{2} \end{equation}\]

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Key Concepts

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

factorization
When simplifying radicals, the first important step is factorization. This means breaking down the number inside the square root into its prime factors. Prime factors are numbers that cannot be divided any further, except by 1 and themselves.
For example, in the exercise, we are simplifying \(\root{18}\). We start by factorizing 18. We can write 18 as a product of its prime numbers:
  • First, 18 can be divided by 2 to get 9.
  • Then, 9 can be divided by 3 to get 3.
This gives us the prime factors: 18 = 2 脳 3 脳 3.
The factorization step is crucial as it sets up the next stages of simplification.
perfect squares
The next concept to understand is perfect squares. A perfect square is a number that can be expressed as the product of an integer with itself. For instance, 1, 4, 9, 16 are all perfect squares, because:
  • 1 = 1 脳 1
  • 4 = 2 脳 2
  • 9 = 3 脳 3
  • 16 = 4 脳 4
Recognizing perfect squares within your factors makes the simplification process easier.
In our example, after factorizing 18 to 2 脳 3 脳 3, we identify the perfect square within these factors. We see that 3 脳 3 = 9, and 9 is a perfect square. This leads us to the next step of separating the perfect square from the remaining factors.
square roots
To simplify a radical fully, you need to deal with square roots. The square root of a perfect square is always an integer. For example,
  • The square root of 9 is 3, since 3 脳 3 = 9.
  • Similarly, the square root of 16 is 4, because 4 脳 4 = 16.
In our given exercise, we separate 18 into \(\root{9 \times 2}\).
We can then take the square root of the perfect square 9, resulting in 3.
This simplifies the expression to: \(\root{18} = 3 \root{2}\).
By dealing with the perfect square, we make our final expression cleaner and easier to manage.

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