Problem 99 - Fundamentals of Mathematics

Question

For the following problems, find each value. Reduce answers to lowest terms or convert to mixed numbers. $$\sqrt{\frac{16}{25}}$$

Step-by-Step Solution

Answer

\( \sqrt{\frac{16}{25}} = \frac{4}{5} \).

Step 1: Understanding the Problem

The problem requires us to find the value of the square root of a fraction \( \frac{16}{25} \). Then, we need to simplify the answer.

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

To find the square root of a fraction \( \frac{16}{25} \), we need to apply the square root separately to the numerator and the denominator. This gives us: \( \sqrt{\frac{16}{25}} = \frac{\sqrt{16}}{\sqrt{25}} \).

Step 3: Calculate the Square Roots

Calculate \( \sqrt{16} \) and \( \sqrt{25} \). Since \( 16 \) is a perfect square, \( \sqrt{16} = 4 \). Similarly, since \( 25 \) is a perfect square, \( \sqrt{25} = 5 \).

Step 4: Simplify the Fraction

Replace the square roots from Step 3 into the fraction: \( \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \). The fraction \( \frac{4}{5} \) is already in its lowest terms.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, often used to simplify expressions and solve equations. When you see the square root symbol, it means you are looking for a number that, when multiplied by itself, gives the original number beneath the root. For example, the square root of 16 is 4, because 4 times 4 equals 16.
In terms of fractions, if you have a square root applied to the entire fraction, like in \( \sqrt{\frac{16}{25}} \), you can take the square root of the top (numerator) and the bottom (denominator) separately. This approach greatly simplifies finding the square root of a fraction.

• The square root of a fraction like \( \frac{16}{25} \) is \( \frac{\sqrt{16}}{\sqrt{25}} \).
• Each part of the fraction has its square root calculated independently.

This method not only breaks the task into smaller, more manageable parts but also ensures accuracy in calculation.

Perfect Squares

Perfect squares are integers that are the square of an integer, which simplifies the process of finding their square roots. Examples include numbers like 1, 4, 9, 16, and 25.
A general trait of perfect squares is that their square roots are always whole numbers or integers. Knowing that 16 and 25 are perfect squares is crucial in simplifying \( \sqrt{\frac{16}{25}} \).

• The square root of 16 results in 4, because 4 times 4 equals 16.
• Similarly, the square root of 25 is 5, because 5 times 5 equals 25.

Recognizing perfect squares helps you skip additional steps of trial and error, as you can confidently state the result without intermediate calculations.

Fractions

Fractions represent parts of a whole and consist of two components—numerators and denominators. A fraction like \( \frac{16}{25} \) describes a part of something that is divided into 25 equally sized pieces, where 16 of those pieces are considered.
When simplifying a fraction such as \( \frac{\sqrt{16}}{\sqrt{25}} \), you maintain the structure of a fraction throughout the problem. This involves simplifying each part individually by breaking it into smaller steps, which makes large problems feel more intimate and easy to handle.

Numerators and Denominators

Numerators and denominators are the backbone of fractions, defining both what is being measured and the scale of measurement. The numerator, or the top number in a fraction, indicates how many parts you have. The denominator, at the bottom, tells you into how many parts the whole is divided.
In \( \frac{16}{25} \), the numerator is 16, reflecting the parts you are working with, and the denominator is 25, indicating the total possible parts. Calculating square roots requires applying this concept of division accurately. For instance, when you simplify \( \frac{\sqrt{16}}{\sqrt{25}} \), you handle numerators and denominators separately:

• Find \( \sqrt{16} = 4 \).
• Find \( \sqrt{25} = 5 \).

This leads to a simplified fraction of \( \frac{4}{5} \), neatly solved by focusing on each component in isolation.