Problem 28 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \sqrt{\frac{11}{25}} $$

Step-by-Step Solution

Answer

Answer: The simplified square root of the fraction 11/25 is $\frac{\sqrt{11}}{5}$.

Step 1: Identify prime factors of numerator and denominator

Identify the prime factors of the numerator (11) and the denominator (25). For 11, it is a prime number so it has no other factors except 1 and 11. Therefore: $$ 11 = 11 $$ For 25, it can be factored into 5 * 5. Therefore: $$ 25 = 5 * 5 $$ So the expression can be rewritten as: $$ \sqrt{\frac{11}{5 * 5}} $$

Step 2: Find the square root using prime factors

Now, we'll find the square root of the fraction, using the prime factors: $$ \sqrt{\frac{11}{5 * 5}} = \sqrt{\frac{11}{5^2}} $$

Step 3: Simplify the obtained expression

Now that we have the square root in a simpler form, we can simplify it further: $$ \sqrt{\frac{11}{5^2}} = \frac{\sqrt{11}}{\sqrt{5^2}} = \frac{\sqrt{11}}{5} $$ So the simplified expression is: $$ \frac{\sqrt{11}}{5} $$

Key Concepts

Prime Factorization

Prime factorization is a technique used to break down a number into its prime factors, which are the basic building blocks of that number. The process involves dividing the number by the smallest prime number until it cannot be divided any further. For example, to find the prime factorization of 25, we divide it by 5, which is the smallest prime number that can exactly divide 25, resulting in the factors of 5 multiplied by itself (5 * 5 or 5²).

Prime factorization makes it easier to simplify radical expressions or fractions involving square roots. It allows us to clearly see which numbers can be taken out of the radical as their own separate square root, simplifying the original expression.

Radical Expressions

A radical expression involves roots, and the most common type is the square root. When simplifying radical expressions, such as the square root of a fraction, the goal is to simplify the expression to its most basic form. This can often be accomplished by prime factorization, as seen in our example where we factorize the denominator 25 into 5 * 5.

The square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator, and if the denominator is a perfect square (as 5 * 5 is), it can be simplified further. After prime factorizing both the numerator and the denominator, we are left with an expression that can often be expressed in a more simplified radical form, ensuring it is as concise and as clear as possible.

Simplify Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common factors except 1. This is often achieved by dividing both the numerator and the denominator by their greatest common divisor. However, when fractions involve square roots in their numerator or denominator, simplifying can also involve expressing the fraction with a rational denominator.

To accomplish this with radical expressions, you want to separate the radicals from the fraction when possible, as in the last step of our original problem. By taking the square root of the denominator, which was a perfect square (5²), we were able to place it outside of the radical, leaving the fraction with a rational denominator. This method of simplification is particularly handy as it allows for clearer understanding and easier manipulation of the fractions in later calculations or algebraic expressions.