Problem 9 - College Algebra with Modeling and Visualization

Question

Find the principal square root of the number. Approximate your answer to the nearest hundredth whenever appropriate. $$ 23 $$

Step-by-Step Solution

Answer

The principal square root of 23 is approximately 4.80.

Step 1: Understanding the Principal Square Root

The principal square root of a number is its non-negative square root. For a number \(a\), its principal square root is denoted by \(\sqrt{a}\). This means we are seeking the positive value \(b\) such that \(b^2 = a\).

Step 2: Estimating the Square Root

We need to find \(\sqrt{23}\). Since \(23\) is not a perfect square, we will have to approximate it. We know that \(\sqrt{16} = 4\) and \(\sqrt{25} = 5\), so \(\sqrt{23}\) is between 4 and 5.

Step 3: Approximating using a Calculator

Using a calculator to find \(\sqrt{23}\), we obtain a value of approximately \(4.7958\).

Step 4: Rounding to the Nearest Hundredth

Rounding \(4.7958\) to the nearest hundredth, we look at the thousandth place (which is 5). Since it is 5 or greater, we round up the hundredth place. Thus, \(\sqrt{23} \approx 4.80\).

Key Concepts

Principal Square Root

The principal square root of a number is its non-negative counterpart, meaning it's always positive or zero. When we talk about square roots, we are essentially looking for a number that, when multiplied by itself, gives us the original number. For example, the principal square root of 23 is represented by \( \sqrt{23} \). This square root gives us a positive value such that when it is squared, it results in 23. Understanding which square root to use is vital because every positive number actually has two square roots: one positive (the principal) and one negative. However, in most cases, especially in mathematical problems aimed at beginners, we focus on the principal square root.

Approximation

When a number doesn't have a perfect square root, we need to approximate. This is the case with 23 since it falls between the perfect squares of 16 and 25. Therefore, \( \sqrt{23} \) is between 4 and 5. To find an accurate estimate, we use tools like calculators, which are designed to provide numerical solutions to these non-perfect squares. Calculators might give you a long decimal like 4.7958 when asked for \( \sqrt{23} \). Such approximations are extremely useful in problems where precise calculations are not feasible. They allow us to work with slightly simplified versions of numbers that still provide a good depiction of the actual values.

Rounding Numbers

Rounding is an important skill that simplifies numbers to make them easier to use in everyday life. When you're rounding a number, you change it to make it less exact but quicker to work with. In the context of square roots, once you've approximated a number like \( \sqrt{23} \) to 4.7958, you may need to round to a specific decimal place for clarity or simplicity. In this case, to round to the nearest hundredth, you'd look at the thousandth place digit (5 in this instance). Because it's 5 or higher, we round up. Therefore, \( 4.7958 \) becomes \( 4.80 \). This method provides a neat and concise way to present numbers.