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Chapter 15: Problem 23

Simplify. $$\sqrt{45}$$

Short Answer

Expert verified
\(\sqrt{45} = 3\sqrt{5}\)

Step by step solution

01

Factorize the number

Firstly, identify the factors of 45. The number 45 can be factored into \(9 \times 5\).
02

Identify the perfect square

From the factors, 9 is a perfect square because it's square root is a whole number; \(\sqrt{9} = 3\).
03

Simplify the square root

Apply the property of square roots to simplify \(\sqrt{45}\) to \(\sqrt{9} \times \sqrt{5}\) . Since \(\sqrt{9} = 3\), the final simplified square root is \(3\sqrt{5}\).

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Key Concepts

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

Factorization
When simplifying square roots, factorization is an essential first step. This process involves breaking down a number into its basic building blocks, known as factors. For example, let's take the number 45. A good starting point is to find easy factors such as small prime numbers: 2, 3, 5, etc.
45 is an odd number, so 2 is not a factor. Check next with 3. Dividing 45 by 3, we get 15. Repeating for 15, dividing it by 3 gives 5. So, 45 can be factorized into 3 times itself once and then multiplied by 5.
This means:
  • Original number: 45
  • Factorized form: 3 脳 3 脳 5 (or written as \(3^2 \times 5\))
Factorizing helps in finding perfect squares within the number, which is the next step in simplifying square roots.
Perfect Square Identification
Once factorization is complete, the next step is to identify any perfect squares among the factors. A perfect square is a number that can be expressed as the square of an integer.
Returning to our factorized number, 3 脳 3 脳 5, observe that 3 脳 3 equals 9, which we recognize as a perfect square because its square root is a whole number (3).
This recognition aids in simplifying the original square root because each perfect square can be simplified further:
  • In our example, we have \( \sqrt{9} = 3\)
Identifying perfect squares helps us to reduce the number under the square root sign, making calculations more straightforward.
Square Root Properties
Understanding the properties of square roots allows us to simplify expressions efficiently. A fundamental property states that the square root of a product equals the product of the square roots.
This property is expressed as:
  • \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
For \(\sqrt{45}\), by applying this property, you can express it as \(\sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}\). Then, using the perfect square identified earlier \(\sqrt{9} = 3\), further simplify the expression to \(3\sqrt{5}\).
Using square root properties not only simplifies the calculation but also enhances understanding by showing how numbers and their roots relate.

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