Problem 10 - Algebra 1: Concepts and Skills

Question

Determine whether each expression is rational or irrational. $$ \sqrt{100} $$

Step-by-Step Solution

Answer

The square root of 100 is a rational number.

Step 1: Understanding the number to be squared rooted

The number to be square rooted is 100. So, it needs to be determined whether the square root of 100 is rational or irrational.

Step 2: Calculate the square root

Calculate the square root of 100 by using a scientific calculator or by knowing that any number multiplied by itself will give the square of that number. The square root of 100 is 10. \( \sqrt{100} = 10 \)

Step 3: Determine whether the result is rational or irrational

A rational number can be expressed as a ratio of two integers. Since 10 can be expressed as a ratio of two integers ( 10/1 for example ), it is a rational number. Therefore, the square root of 100 is a rational number.

Key Concepts

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 100 is 10, because multiplying 10 by itself (\( 10 \times 10 \) yields 100. This concept is fundamental in understanding various mathematical and real-world applications, such as calculating areas of squares.

It's helpful to recognize that some square roots result in whole numbers, as seen with the square root of 100 (\( \sqrt{100} = 10 \)), while others do not, such as the square root of 2. Recognizing which numbers have whole number square roots can significantly simplify the process of identifying whether an expression is rational or irrational.

Simplifying Square Roots

Simplifying square roots involves finding the prime factorization of a number and pairing the primes to pull them outside of the square root symbol. For square roots that are already whole numbers, this process is unnecessary; the result is immediately clear. However, for square roots of numbers that are not perfect squares, simplifying can clarify whether the root is rational or irrational. For instance, \( \sqrt{50} \) can be simplified to \( 5\sqrt{2} \) because 50 can be expressed as \( 25 \times 2 \) and \( \sqrt{25} \) is 5. Simplification can provide insight into the nature of the number but does not change its value--only its form.

Properties of Rational Numbers

Rational numbers are numbers that can be represented as a fraction of two integers, where the denominator is not zero. They have a few distinct properties worth noting:

• Closure: The sum, difference, or product of any two rational numbers is also a rational number.
• Existence of Inverses: For every rational number except zero, there exists a multiplicative inverse (reciprocal), which is also a rational number.
• Repeating or Terminating Decimal: Rational numbers in decimal form either terminate after a finite number of digits or repeat a sequence of digits indefinitely.
• Density: Between any two rational numbers, there is another rational number. This shows that they are densely packed along the number line.

Using these properties can help identify whether a number is rational or irrational. For the square root of 100, which is 10, it is clear that it fits the characteristics of rational numbers, such as being able to be expressed as \( \frac{10}{1} \), a fraction of integers.