Problem 52 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{3}{21}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt{\frac{3}{21}}\) is \(\frac{\sqrt{7}}{7}\).

Step 1: Simplify the Fraction

Before taking the square root, simplify the fraction. The fraction \(\frac{3}{21}\) can be simplified because 3 is a factor of 21. Divide both the numerator and the denominator by 3 to get the simplified fraction which is \(\frac{3}{21} = \frac{1}{7}\).

Step 2: Apply the Square Root

Now that \(\frac{3}{21}\) has been simplified to \(\frac{1}{7}\), the next step is to apply the square root to the simplified fraction \(\sqrt{\frac{1}{7}}\). In this case, the square root of 1 is 1, and the square root of 7 remains as is since 7 is a prime number, therefore \(\sqrt{\frac{1}{7}} = \frac{1}{\sqrt{7}}\).

Step 3: Rationalize the Denominator

The final step to simplify this expression is to rationalize the denominator, which means eliminating the square root from the denominator. This can be done by multiplying both the numerator and denominator by \(\sqrt{7}\). Doing so gives: \(\frac{1}{\sqrt{7}}*\frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{7}\).

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. This can make the expressions easier to work with, especially when performing additional operations like taking square roots or solving equations.
To simplify a fraction like \( \frac{3}{21} \):

• First, identify the greatest common factor (GCF) of the numerator and the denominator.
• Here, both 3 and 21 are divisible by 3. Therefore, 3 is the GCF.
• Divide both the numerator and the denominator by this GCF. This gives us \( \frac{3}{21} = \frac{1}{7} \).

The fraction \( \frac{1}{7} \) cannot be reduced further, as 1 and 7 are coprime (i.e., their only common factor is 1). Simplifying fractions is an important skill because it reduces complexity in expressions and calculations.

Rationalizing the Denominator

Rationalizing the denominator involves rewriting a fraction so that there are no irrational numbers (like square roots) in the denominator. This is a standard mathematical practice because it is often easier to work with expressions without irrational denominators.
Let’s look at how you would rationalize a denominator, using \( \frac{1}{\sqrt{7}} \) as an example.

• You start by multiplying both the numerator and the denominator by the square root in the denominator. In this case, that's \( \sqrt{7} \).
• Thus, you have \( \frac{1}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{\sqrt{7}}{7} \).
• Notice how multiplying by \( \frac{\sqrt{7}}{\sqrt{7}} \), which is equivalent to 1, doesn't change the value of the expression but shifts the square root from the denominator to the numerator, removing the irrational part from the denominator.

Remembering to rationalize helps maintain clarity and simplicity in mathematical expressions, especially when dealing with more complex calculations.

Prime Numbers

Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves. Understanding prime numbers is crucial in simplifying square roots and fractions, as they often determine when a number can or cannot be divided further. Prime numbers are essential in a wide range of mathematical problems.
For instance, in simplifying square roots, knowing a number is prime tells you it cannot be factored further.
Looking at our example, 7 is a prime number, which is why \( \sqrt{7} \) remains as is instead of being simplified further. When the square root of a non-prime number is taken, it often leads to simpler forms if the number can be decomposed into squares of other numbers.
Prime numbers include: 2, 3, 5, 7, 11, 13, and so on. They act as the basic building blocks of numbers, and their properties play a significant role in various areas of mathematics, such as cryptography and number theory.