Problem 4
Question
Fill in the blanks. The denominator of the fraction \(\frac{4}{\sqrt{5}}\) is an __________ number.
Step-by-Step Solution
Verified Answer
irrational
1Step 1: Identify the denominator
The given fraction is \(\frac{4}{\sqrt{5}}\). The denominator is \(\sqrt{5}\).
2Step 2: Determine the type of number
\(\sqrt{5}\) is a square root of a non-perfect square number, so it is not a rational number but an irrational number.
3Step 3: Fill in the blank
Given the identification from Step 2, the blank can be filled with the word 'irrational'.
Key Concepts
square rootsfractionsrational numbers
square roots
Square roots are a fundamental concept in mathematics, helping us determine a number that, when multiplied by itself, results in the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 equals 9. The symbol used to represent square roots is the radical sign (√). It's essential to note that not every number has a neat, whole number square root.
Non-perfect squares, like 5, have square roots that are not integers. In these cases, the square root is considered irrational, meaning it cannot be exactly expressed as a simple fraction. When dealing with square roots in fractions, such as in \(\frac{4}{\sqrt{5}}\), it's important to recognize that simplifying or manipulating these expressions may result in irrational numbers.
Non-perfect squares, like 5, have square roots that are not integers. In these cases, the square root is considered irrational, meaning it cannot be exactly expressed as a simple fraction. When dealing with square roots in fractions, such as in \(\frac{4}{\sqrt{5}}\), it's important to recognize that simplifying or manipulating these expressions may result in irrational numbers.
fractions
Fractions represent a part of a whole and consist of two numbers: a numerator and a denominator. The numerator is the top part, showing how many parts we have. The denominator is the bottom part, indicating the total number of equal parts the whole is divided into. For example, in \(\frac{4}{\sqrt{5}}\), 4 is the numerator, and \(\sqrt{5}\) is the denominator.
Fractions can be simplified or altered by multiplying or dividing the numerator and the denominator by the same number, without changing the value of the fraction. When the denominator of a fraction is an irrational number (like \(\sqrt{5}\)), it can sometimes be helpful to rationalize the denominator by multiplying both the numerator and the denominator by a suitable value that makes the new denominator a rational number. This technique aids in making the expression more comfortable to work with.
Fractions can be simplified or altered by multiplying or dividing the numerator and the denominator by the same number, without changing the value of the fraction. When the denominator of a fraction is an irrational number (like \(\sqrt{5}\)), it can sometimes be helpful to rationalize the denominator by multiplying both the numerator and the denominator by a suitable value that makes the new denominator a rational number. This technique aids in making the expression more comfortable to work with.
rational numbers
Rational numbers are any numbers that can be expressed as a fraction with both the numerator and the denominator as integers, where the denominator is not zero. For example, \(\frac{3}{4}\) and \(\frac{-2}{7}\) are rational numbers because they can be written as simple fractions of integers.
In contrast, irrational numbers cannot be written as simple fractions. They have non-repeating, non-terminating decimal expansions. Examples include \(\pi\) and \(\sqrt{2}\). When working with fractions like \(\frac{4}{\sqrt{5}}\), recognizing the type of number in the denominator is crucial. Since \(\sqrt{5}\) is irrational, the entire fraction represents a value that isn't neatly expressible as a single fraction with integer components.
In contrast, irrational numbers cannot be written as simple fractions. They have non-repeating, non-terminating decimal expansions. Examples include \(\pi\) and \(\sqrt{2}\). When working with fractions like \(\frac{4}{\sqrt{5}}\), recognizing the type of number in the denominator is crucial. Since \(\sqrt{5}\) is irrational, the entire fraction represents a value that isn't neatly expressible as a single fraction with integer components.
Other exercises in this chapter
Problem 4
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Fill in the blanks. To _____ \(\sqrt{24}\) means to write it as \(2 \sqrt{6}.\)
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