Problem 10 - Elementary Algebra

Question

Simplify each square root. $$ \sqrt{\frac{2}{7}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the given square root is: $$\frac{\sqrt{2}}{\sqrt{7}}$$.

Step 1: Separate the square root of the numerator and the denominator

Separate the square root into the numerator and the denominator as follows: $$ \sqrt{\frac{2}{7}} = \frac{\sqrt{2}}{\sqrt{7}} $$

Step 2: Simplify the square roots, if possible

In this case, neither the numerator nor the denominator can be further simplified in terms of square roots, as 2 and 7 are both prime numbers. Therefore, the given expression is already in its simplest form: $$ \frac{\sqrt{2}}{\sqrt{7}} $$ So, the simplified form of the given square root is: $$ \sqrt{\frac{2}{7}} = \frac{\sqrt{2}}{\sqrt{7}} $$

Key Concepts

Prime Numbers

Prime numbers are special kinds of numbers because they have only two factors: 1 and themselves. This means that you cannot divide a prime number by any other number without leaving a remainder.
For example, 2, 3, 5, 7, 11, 13, and so on are prime numbers. Here are a few key points about them:

• Prime numbers start from 2, which is the first and smallest prime number.
• They are crucial in various areas of mathematics due to their indivisibility.
• In the context of simplifying square roots, knowing if a number is prime helps us understand that it cannot be broken down further into simpler factors.

In our exercise, 2 and 7 are both primes, which is why \( \sqrt{2} \) and \( \sqrt{7} \) are already in their simplest forms. Recognizing prime numbers is especially useful when simplifying fractions or expressions, allowing you to see at a glance that no further simplification is possible.

Numerator and Denominator

In any fraction, you have a numerator and a denominator. The numerator is the number above the division line, and the denominator is the number below it. These two parts serve different purposes:

• Numerator: Indicates how many parts we have out of a whole or a set.
• Denominator: Tells us the total number of equal parts the whole is divided into.

For example, within the fraction \( \frac{2}{7} \), 2 is the numerator and 7 is the denominator. Understanding these components is critical when working with fractions, especially when involving operations like addition or subtraction.
In the context of our exercise, understanding where to place the square roots, i.e., \( \frac{\sqrt{2}}{\sqrt{7}} \), is essential in maintaining the integrity of the fraction.

Square Root Properties

Square roots have interesting properties that allow us to manipulate and simplify them in various math problems. When dealing with square roots, it's important to remember:

• The square root of a number is a value that, when multiplied by itself, gives the original number.
• The square root operation involves both the numerator and the denominator separately if the expression is a fraction.
• \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \), which means you can multiply the numbers inside the square root first, then simplify.

In our original problem, \( \sqrt{\frac{2}{7}} = \frac{\sqrt{2}}{\sqrt{7}} \) shows the property where the square root of a fraction is handled by taking the square root of each part—numerator and denominator—individually. Recognizing and applying these properties can be extremely helpful in making expressions easier to work with and understand.