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Chapter 7: Problem 407

Is \(\sqrt{144}\) rational or irrational?

Short Answer

Expert verified
The square root of 144 is rational because it equals 12, an integer.

Step by step solution

01

Understand the Definition

A rational number can be expressed as the quotient of two integers, i.e., \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \) is not zero. An irrational number cannot be expressed this way.
02

Calculate the Square Root

Determine the square root of 144. \( \sqrt{144} = 12 \).
03

Verify the Result

Check if the result is an integer. The square root of 144 is 12, which is an integer.
04

Conclusion

Since 12 can be expressed as \( \frac{12}{1} \), it is a rational number. Thus, \( \sqrt{144} \) is rational.

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Key Concepts

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

Understanding Square Roots
Square roots are numbers that produce a specified number when multiplied by themselves. For instance, the square root of 25 is 5, because 5 multiplied by itself equals 25. Square roots can be whole numbers (perfect squares) or decimals. When a number is a perfect square, like 144, its square root is an integer.
What Are Integers?
Integers are whole numbers that can be positive, negative, or zero. They do not include fractions or decimals. For example, -3, 0, and 7 are all integers. In our exercise, we calculated the square root of 144 and arrived at 12, which is an integer.
Rational vs Irrational Numbers
Rational numbers can be expressed as a fraction \left( \frac{a}{b} \right) where both \(a\) and \(b\) are integers, and \(b\) is not zero. Examples include 1/2, 4, and -2/3. Irrational numbers, on the other hand, cannot be expressed as such fractions. They are non-repeating, non-terminating decimals, like \sqrt{2} and \pi. From our example, since 12 is an integer, it can be expressed as \left( \frac{12}{1} \right). Therefore, 12 is a rational number.

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Most popular questions from this chapter

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