Problem 45 - Elementary Algebra

Question

\(\sqrt{15}\)

Step-by-Step Solution

Answer

About 3.872

Step 1: Identify the Problem

The task is to find the square root of the number 15, which is represented as \(\root{15}\).

Step 2: Estimate the Square Root

Determine two perfect squares between which the number 15 lies. These numbers are 9 and 16, which have square roots of 3 and 4, respectively.

Step 3: Refine the Estimate

Since \(\sqrt{15}\) lies between \(\sqrt{9} = 3\) and \(\sqrt{16} = 4\), estimate that \(3.8^2 = 14.44\) and \(3.9^2 = 15.21\). Given 15 is closer to 15.21, \(\sqrt{15} \approx 3.9\).

Step 4: Use a Calculator for Precision

Use a calculator to find a more accurate value of \(\sqrt{15}\). The calculator shows that \(\sqrt{15} \approx 3.872\).

Key Concepts

Square Roots

Square roots are one of the fundamental concepts in mathematics. When you take the square root of a number, you are finding a value that, when multiplied by itself, gives you the original number. For instance, the square root of 9 is 3 because 3 multiplied by 3 equals 9. The symbol for square root is \(\root{}\). In our given example, we need to find \(\root{15}\).

To estimate square roots, you can identify two whole numbers between which the square root lies. This helps provide an initial rough estimate which we later refine.

Perfect Squares

Perfect squares play a vital role in estimating square roots. A perfect square is an integer that is the square of another integer. For example, 1, 4, 9, 16, and 25 are perfect squares because they are 1\(^2\), 2\(^2\), 3\(^2\), 4\(^2\), and 5\(^2\), respectively.

In our exercise of finding \(\root{15}\), we observe that 15 lies between two perfect squares, 9 (which is 3\(^2\)) and 16 (which is 4\(^2\)). Knowing these boundaries helps us estimate that \(\root{9} = 3\) and \(\root{16} = 4\). Therefore, \(\root{15}\) should be somewhere between 3 and 4.

This step is crucial for making more precise approximations and understanding the relationship between numbers.

Approximation

Approximating square roots helps provide a more practical value when an exact number isn't needed or is difficult to find. After determining that \(\root{15}\) is between 3 and 4, we can narrow it down further.

We try values like 3.8 and 3.9 by squaring them. Squaring 3.8 gives 14.44, and squaring 3.9 provides 15.21. Comparing these, we see that 15 is closer to 15.21, implying \(\root{15}\) is roughly 3.9.

For even greater accuracy, a calculator can be used, which shows \(\root{15}\) to be approximately 3.872. Approximations are useful in real-world applications where exact values are often unnecessary.