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Chapter 1: Problem 293

In the following exercises, simplify. $$ \sqrt{144} $$

Short Answer

Expert verified
The square root of 144 is 12.

Step by step solution

01

Understand the square root operation

The square root operation asks for a number which, when multiplied by itself, gives the original number under the square root. In this case, we need to find a number that, when squared, results in 144.
02

Identify the perfect square

Recognize that 144 is a perfect square. This means there exists an integer whose square equals 144.
03

Determine the root

The number that multiplied by itself equals 144 is 12, because \[ 12 \times 12 = 144 \]
04

Write the simplified form

Thus, the square root of 144 is: \[ \sqrt{144} = 12 \]

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Key Concepts

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

square root operation
The square root operation is a mathematical function that seeks to find a number which, when multiplied by itself, gives the original number under the square root symbol. In simpler terms, if you have a square with an area of 144 square units, the side length of that square is the square root of 144.
In our example, we need to find a number that, when squared (multiplied by itself), equals 144. This means we seek a value such that \(\text{a}^2 = 144\).
perfect square
A perfect square is a number that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, and 25 are all perfect squares since they can be expressed as \(\text{1}^2, \text{2}^2, \text{3}^2, \text{4}^2, \text{5}^2\) respectively.
In this case, 144 is identified as a perfect square because it equals \(\text{12} \times 12\). Recognizing perfect squares can make simplifying square roots much easier as it immediately points us to an integer root.
simplification
Simplification in the context of square roots involves finding the simplest form of the expression. This is often done by determining if the number under the square root is a perfect square. If it is, you can simplify the square root by determining the integer whose square is the given number.
In our example, simplifying \(\text{sqrt}(144)\) involves identifying that 144 is a perfect square and then finding the integer root, which is 12. Thus, \(\text{sqrt}(144) = 12\).
integer roots
Integer roots refer to whole numbers that, when multiplied by themselves, result in the original number under the square root. Only perfect squares have integer roots.
For example, the integer root of 144 is 12 because \(\text{12} \times \text{12} = 144\). When dealing with square roots, if the number under the root isn't a perfect square, the root won't be an integer. However, for perfect squares like 144, it's straightforward: \(\text{sqrt}(144) = 12\).

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