Problem 679 Identify whether each number is ... [FREE SOLUTION]

Chapter 1: Problem 679

Identify whether each number is rational or irrational. $$ \text { (a) } \sqrt{25} \text { (b) } \sqrt{30} $$

Short Answer

Expert verified

(a) Rational (b) Irrational

Step by step solution

Understand the Problem

Determine if each of the given numbers is rational or irrational. A rational number can be expressed as a fraction of two integers, while an irrational number cannot.

Analyze Part (a)

Examine $ \sqrt{25} $. Calculate the square root of 25. Since 25 is a perfect square, $ \sqrt{25} = 5 $. The result is a whole number, meaning it is rational because it can be expressed as $ \frac{5}{1} $.

Analyze Part (b)

Examine $ \sqrt{30} $. Calculate the square root of 30. Since 30 is not a perfect square, the square root of 30 is not a whole number and cannot be expressed as a fraction of two integers, making it an irrational number.

Summarize the Results

Based on the analysis, $ \sqrt{25} $ is rational and $ \sqrt{30} $ is irrational.

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Key Concepts

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

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. For example, 25 is a perfect square because it equals $5 \times 5$.

Common perfect squares include:

4 (since $2 \times 2 = 4$)
9 (since $3 \times 3 = 9$)
16 (since $4 \times 4 = 16$)

Identifying whether a number is a perfect square is helpful when determining if its square root is rational.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 25 is 5, because $5 \times 5 = 25$.

If a number is a perfect square, its square root will be an integer. This makes understanding square roots essential when deciding if a number is rational or irrational.

Integers

Integers are whole numbers that can be positive, negative, or zero. Examples include -3, 0, and 7.

In the context of rational and irrational numbers, knowing whether a square root results in an integer can help determine the rationality of a number. For example, since $ \sqrt{25} = 5 $, and 5 is an integer, $ \sqrt{25} $ is rational.

Fraction

A fraction represents a part of a whole and is expressed as $\frac{a}{b}$, where $a$ and $b$ are integers and $b eq 0$.

Rational numbers can be expressed as simple fractions, like $\frac{3}{4}$ or $\frac{5}{1}$. On the other hand, irrational numbers cannot be written as such fractions. For example, since 30 is not a perfect square, $\sqrt{30}$ is irrational and cannot be expressed as a fraction of two integers.

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91影视

Identify whether each number is rational or irrational. $$ \text { (a) } \sqrt{25} \text { (b) } \sqrt{30} $$

Short Answer

Step by step solution

Understand the Problem

Analyze Part (a)

Analyze Part (b)

Summarize the Results

Key Concepts

Perfect Squares

Square Roots

Integers

Fraction

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