Problem 6 - Advanced Calculus

Question

(a) Which is larger, \([\sqrt{243 / 3}]\) or \([12 / \sqrt{5}]\) ? (b) Find a rational number between \(\sqrt{37}\) and \(\sqrt{39}\).

Step-by-Step Solution

Answer

(a) \([12 / \sqrt{5}]\) is larger. (b) One possible rational number between \(\sqrt{37}\) and \(\sqrt{39}\) is 6.5. Other rational numbers in this range could also be chosen.

Step 1: Simplify the expressions

First, we simplify both \([\sqrt{243 / 3}]\) and \([12 / \sqrt{5}]\). \([\sqrt{243 / 3}]\) simplifies to 9. \([12 / \sqrt{5}]\) simplifies to \(12*\sqrt{5} / 5\) which is roughly 10.68.

Step 2: Compare the Simplified expressions

Now that they are simplified, it becomes clear that \([\sqrt{243 / 3}]\) is less than \([12 / \sqrt{5}]\), as 9 is less than 10.68. Therefore, \([12 / \sqrt{5}]\) is larger.

Step 3: Check the integers between the square roots in part b

By checking the integers between the two given square roots, we notice that 6 and 7 are between \(\sqrt{37}\) (approximately 6.08) and \(\sqrt{39}\) (approximately 6.24). So we can choose a rational number between these two roots.

Step 4: Choose a rational number

Any rational number between 6 and 7 is acceptable. For example, the simple and obvious rational number would be 6.5.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. These numbers play an important role in mathematics, as they allow us to express exact values like middle points or averages.

• For instance, the rational number 6.5 can easily be expressed as \( \frac{13}{2} \).
• Rational numbers feel like familiar friends because they include a host of numbers we know well, like all integers and simple fractions.

In problems involving square roots, finding a rational number between two roots helps give a clearer understanding. If you have two roots such as \( \sqrt{37} \) and \( \sqrt{39} \), identifying rational numbers between them, such as 6.5, is a common approach. This is because 6 and 7, which are whole numbers, fit snugly between the intermediate decimal values of the square roots.

Simplifying Radicals

Simplifying radicals involves expressing root expressions in their simplest form, where possible. This step can reveal the true size of the number you are dealing with, making comparisons much more straightforward.

• For example, simplifying \( \sqrt{243 / 3} \) involves dividing 243 by 3, which gives 81. The square root of 81 is 9. Therefore, \( \sqrt{243 / 3} \) simplifies to 9.
• In contrast, to simplify \( \frac{12}{\sqrt{5}} \), it is often rationalized. This involves multiplying by a form of 1 – in this case, \( \frac{\sqrt{5}}{\sqrt{5}} \) – to remove the radical from the denominator.
• This operation turns \( \frac{12}{\sqrt{5}} \) into \( \frac{12 \cdot \sqrt{5}}{5} \), approximately equal to 10.68.

These simplifications and transformations are crucial because they allow for easy comparison of magnitudes. When each expression is reduced to a simpler form, assessing which is larger or fits within a particular range becomes clearer.

Square Roots

The concept of square roots is foundational in math. A square root of a number is a value that, when multiplied by itself, gives the original number. Every positive number has two square roots, one positive and one negative; however, in most practical applications, we use the positive counterpart.

• For instance, the square root of 39 is somewhere between 6 and 7 because \( 6^2 = 36 \) and \( 7^2 = 49 \).
• Square roots help reveal approximate values of larger numbers and fit a continuous number spectrum between integers.

Specifically for our exercise, understanding that \( \sqrt{37} \) and \( \sqrt{39} \) are between two integers aids in recognizing valid rational numbers in that gap. Square roots can often lead to irrational numbers, but rational approximations are useful for many applications.