Problem 10 - Algebra A Combined Function

Question

Find each square root. Assume that all variables represent nonnegative real numbers. $$ \sqrt{\frac{9}{25}} $$

Step-by-Step Solution

Answer

The square root is \( \frac{3}{5} \).

Step 1: Understand the Problem

We need to find the square root of the fraction \( \frac{9}{25} \). Given that the variables represent nonnegative real numbers, we can look for the principal (positive) square root.

Step 2: Apply the Square Root to the Numerator and Denominator Separately

To find the square root of a fraction, apply the square root to both the numerator and the denominator. That is:\[\sqrt{\frac{9}{25}} = \frac{\sqrt{9}}{\sqrt{25}}\]

Step 3: Calculate the Square Roots

Find the square root of the numerator and the denominator. The square root of \(9\) is \(3\) since \(3^2 = 9\). The square root of \(25\) is \(5\) since \(5^2 = 25\).Thus, we have:\[\frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5}\]

Step 4: Simplify the Fraction

The fraction \(\frac{3}{5}\) is already in its simplest form, as both the numerator and the denominator are coprime (they have no common factors other than 1).

Key Concepts

Fraction Simplification

Fraction simplification is a helpful skill in simplifying complex mathematical expressions. When simplifying a fraction, the goal is to reduce it to its lowest terms so that both the numerator and the denominator are as small as possible while retaining the same value.

• First, identify the greatest common divisor (GCD) of both the numerator and the denominator. This means finding the largest number that divides into both numbers without leaving a remainder.
• Divide both the numerator and the denominator by their GCD to simplify the fraction.
• If the GCD is 1, it means the fraction is already in its simplest form, as is the case of \(\frac{3}{5}\), where 3 and 5 are coprime.

Simplifying fractions can make calculations easier and help us better understand the size or proportion of values in relation to one another.

Nonnegative Real Numbers

Nonnegative real numbers are numbers that are greater than or equal to zero. They include all the positive real numbers, such as 1, 2, 3, and fractions like \(\frac{1}{2}\), as well as zero itself.

• This concept is particularly significant in many mathematical contexts, such as when we're dealing with square roots. The square root operation is only defined for nonnegative numbers, as the square of a real number is never negative.
• Variables representing nonnegative real numbers can often simplify problem-solving because any results computed will be positive or zero, reducing the complexity of handling negative numbers.

Understanding nonnegative real numbers allows us to focus on positive results, especially when calculating principal square roots, where negativity doesn't arise.

Principal Square Root

The principal square root of a nonnegative real number is the nonnegative number that, when squared, yields the original number. It is important to note that when we talk about taking the square root of a number, we generally refer to its principal square root.

• The square root symbol \(\sqrt{}\) traditionally conveys finding the principal square root.
• When finding the square root of fractions like \(\sqrt{\frac{9}{25}}\), the process involves separately taking the principal square root of the numerator and the denominator.
• In our example, the principal square root of 9 is 3, and for 25, it is 5, hence the fraction \(\frac{3}{5}\). This fraction represents the principal square root of the original fraction.

Remembering that the principal square root is the positive root can prevent errors when working with equations and helps in maintaining clarity when solutions need to be nonnegative, particularly with real-world applications.