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Chapter 10: Problem 16

Simplify completely. If the radical is already simplified, then say so. $$\sqrt{15}$$

Short Answer

Expert verified
Since there are no perfect square factors of 15, the square root is already simplified. So, \(\sqrt{15}\) remains as \(\sqrt{15}\).

Step by step solution

01

Find the factors of 15

First, list the factors of 15 to find if there are any perfect square factors. The factors of 15 are: 1, 3, 5, and 15.
02

Identify perfect square factors

Check if any of the factors is a perfect square number: 1, 3, 5, and 15 are not perfect squares since none of them can be written as a square of an integer. (The nearest perfect squares are 4 and 9, that is, \(2^2\) and \(3^2\), respectively. But these aren't factors of 15)
03

State the simplification

Since there are no perfect square factors of 15, the square root is already simplified. So, \(\sqrt{15}\) is simplified and remains as \(\sqrt{15}\).

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Key Concepts

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

Perfect Square Factors
When simplifying radicals, it's important to look for perfect square factors. Perfect squares are numbers that can be expressed as the square of an integer. Examples include 1, 4, 9, 16, 25, and so on. When you find a perfect square as a factor within a radical, it simplifies the task of finding the square root. For example, the number 36 is a perfect square, as it equals to the square of 6.
  • Some other common perfect squares are: 4 (since 4 = 2 x 2), 9 (3 x 3), 16 (4 x 4), etc.
  • Identifying these makes simplification possible by transforming the radical into a simpler form.
In the given exercise, \(\sqrt{15}\), checking for perfect square factors is crucial. Since none of the factors of 15 (1, 3, 5, and 15) are perfect squares, \(\sqrt{15}\) remains unsimplified.
Square Root Simplification
Square root simplification involves breaking down the number under the radical, if possible, into factors, one of which is a perfect square. You do this to simplify the radical expression into a more concise form. This process greatly depends on the integer involved having a perfect square factor. For instance, consider simplifying \(\sqrt{50}\). The factors of 50 are: 1, 2, 5, 10, 25, 50. Notice that 25 is a perfect square (5 x 5). Thus, \(\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\).
  • If a number doesn't have a perfect square factor other than 1, the radical cannot be simplified further.
  • This was the case with \(\sqrt{15}\), since no higher perfect square factors exist other than 1.
The principle here is to always break down the number until you find a perfect square, if possible.
Factors of a Number
Understanding factors is crucial in solving various mathematical problems, including radical simplifications. Factors are numbers you multiply together to get another number. For example, the factors of 15 are 1, 3, 5, and 15. This means these numbers can be multiplied in pairs to result in 15 (1 x 15, 3 x 5).When working with radicals, factoring helps identify if any part of the number can be simplified.
  • If a number has a factor that is also a perfect square, it can simplify the radical expression.
  • If none of the factors are perfect squares, as in the case of 15, the radical cannot be simplified further and remains as it is, \(\sqrt{15}\).
By understanding how to find and use factors, you improve your ability to simplify and manipulate numbers in radical form effectively.

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