Problem 40 - Elementary Algebra

Question

For the following problems, find the two square roots of the given number. $$ \frac{121}{225} $$

Step-by-Step Solution

Answer

Answer: The two square roots of the given fraction are: $\pm \frac{11}{15}$.

Step 1: Simplify the fraction

First, we need to simplify the fraction: $$ \frac{121}{225} $$ Factor the numerator and the denominator to their prime factors. Numerator: $$ 121 = 11^2 $$ Denominator: $$ 225 = 15^2 = (3 \cdot 5)^2 = 3^2 \cdot 5^2 $$ Now we can simplify the fraction: $$ \frac{11^2}{3^2 \cdot 5^2} $$

Step 2: Calculate the square root of the numerator and denominator separately

Next, we find the square roots of the numerator and the denominator: Square root of numerator: $$ \sqrt{11^2} = 11 $$ Square root of denominator: $$ \sqrt{3^2 \cdot 5^2} = \sqrt{3^2} \cdot \sqrt{5^2} = 3 \cdot 5 = 15 $$

Step 3: Write the two possible square roots of the fraction

Finally, we can write the two possible square roots of the given fraction: 1. Positive square root: $$ \sqrt{\frac{11^2}{3^2 \cdot 5^2}} = \frac{\sqrt{11^2}}{\sqrt{3^2 \cdot 5^2}} = \frac{11}{15} $$ 2. Negative square root: $$ -\sqrt{\frac{11^2}{3^2 \cdot 5^2}} = \frac{-\sqrt{11^2}}{\sqrt{3^2 \cdot 5^2}} = -\frac{11}{15} $$ So, the two square roots of the given number are: $$ \pm \frac{11}{15} $$

Key Concepts

Simplify Fractions

Simplifying fractions is a fundamental skill in mathematics that involves reducing a fraction to its most basic form, where the numerator and denominator share no common factors other than 1. In the context of solving square roots of fractions, this preliminary step is crucial.

Take, for example, the fraction \( \frac{121}{225} \). To simplify it, we perform the prime factorization of both the numerator and the denominator. Once we have the prime factors, we look for common factors and cancel them out, eventually simplifying the fraction to its lowest terms.

When dealing with square roots, as in the exercise, simplifying the fraction beforehand can make it easier to take the square root of the numerator and denominator separately, which leads to finding the square roots of the fraction with greater ease.

Prime Factorization

Prime factorization is the process of breaking down a composite number into the product of its prime factors. Understanding this concept is crucial when simplifying fractions or finding the square roots of fractions.

Consider our example, where we factor \(121\) into \(11^2\) and \(225\) into \(3^2 \cdot 5^2\). By representing these numbers as powers of prime numbers, the process of finding the square roots becomes streamlined, since a square root of a perfect square, which is a number that's a square of an integer, gives us a whole number.

This method is also widely used in various branches of mathematics and helps with operations involving radicals and rational exponents.

Radicals and Rational Exponents

The concept of radicals and rational exponents plays a significant role when dealing with square roots, especially of fractions. A radical, often represented by the square root symbol \( \sqrt{\cdot} \), signifies the operation of finding a number that, when multiplied by itself, yields the original number. Rational exponents, on the other hand, express powers as fractions.

For instance, \( \sqrt{11^2} \) simplifies to \( 11 \) because the exponent \( \frac{2}{2} \) (the power of 11 inside the radical over the index of the square root, which is 2) reduces to 1. Here, the laws of exponents are applied to radicals since they can be thought of as exponents to the power of \( \frac{1}{2} \).

The exercise illustrates this when the square roots of the numerator and denominator are calculated separately, and both radicals simplify to integers, further facilitating the computation of the square roots of the whole fraction.

Absolute Value of Real Numbers

The absolute value of a real number is its distance from zero on the number line, disregarding its sign. It ensures that a number is always non-negative. This concept is relevant when discussing the square roots of a number since each non-negative real number actually has two square roots: one positive and one negative.

In the problem \( \frac{121}{225} \), the square root is given as \( \pm \frac{11}{15} \), capturing both the positive and negative roots. This notation acknowledges the absolute value: while \( \frac{11}{15} \) directly represents the positive root, the negative root \( -\frac{11}{15} \) is the same distance from zero but on the opposite side of the number line.

When solving square roots of fractions (or any number), it's important to consider both roots to ensure a complete understanding and solution to the problem.