Problem 57 - Prealgebra

Question

Estimate each square root to the nearest whole number. Do not use a calculator. $$-\sqrt{17.3}$$

Step-by-Step Solution

Answer

\(-\sqrt{17.3}\) is approximately -4.

Step 1: Identify Perfect Squares

First, we need to identify the perfect squares that are closest to 17.3. These are 16 \((4^2)\) and 25 \((5^2)\). Since 17.3 is closer to 16 than to 25, it is logical to guess that \(\sqrt{17.3}\) will be closer to 4.

Step 2: Determine Range

Since we know that \(\sqrt{16} = 4\) and \(\sqrt{25} = 5\), \(\sqrt{17.3}\) must be between 4 and 5. But given 17.3 is much closer to 16, it will be just over 4.

Step 3: Estimate Square Root

Knowing \(\sqrt{17.3}\) is just over 4, we round this up to the nearest whole number. Since 17.3 is closer to 16 than 25, \(\sqrt{17.3}\) is slightly more than 4 and not enough to round up to 5, so \(\sqrt{17.3} \approx 4\).

Step 4: Apply Negative Sign

The original problem includes a negative sign outside the square root, so the final estimation is \(-\sqrt{17.3}\). Therefore, \(-\sqrt{17.3} \approx -4\).

Key Concepts

Perfect Squares

A perfect square is defined as a number that can be expressed as the product of an integer multiplied by itself. For instance, 16 is a perfect square because it is the result of multiplying 4 by 4, or mathematically expressed as \(4 \times 4 = 16\). This same logic applies to 25, which is the result of \(5 \times 5 = 25\). Creating a mental list of perfect squares is extremely helpful when estimating square roots, especially when you do not have a calculator handy. Some useful perfect squares to remember include:

• \(1 \times 1 = 1\)
• \(2 \times 2 = 4\)
• \(3 \times 3 = 9\)
• \(4 \times 4 = 16\)
• \(5 \times 5 = 25\)
• \(6 \times 6 = 36\)
• \(7 \times 7 = 49\)
• \(8 \times 8 = 64\)
• \(9 \times 9 = 81\)
• \(10 \times 10 = 100\)

These benchmarks offer a reference point that can help you determine which perfect squares a given number is close to, allowing you to estimate its square root more effectively.

Estimating Square Roots

Estimating square roots involves finding two perfect squares that “sandwich” the number in question. If our goal is to estimate \(\sqrt{17.3}\), we identify that 17.3 falls between the perfect squares of 16 and 25.Here’s how you can break down the process:

• Recognize that 16 is \(4^2\) and 25 is \(5^2\).
• Observe that 17.3 is closer to 16 than 25, which suggests its square root is closer to 4.

Understanding this relation means that \(\sqrt{17.3}\) falls between 4 and 5, leaning closer to the lower number. This method doesn't give an exact answer but a close estimate. So when faced with problems asking for an estimated square root, always determine which perfect squares are on either side of the given number to narrow down your potential answers.

Whole Numbers

Whole numbers are foundational building blocks in mathematics, consisting of all the non-negative integers, including zero. That is, the set of whole numbers is \( \{0, 1, 2, 3, 4, 5, \ldots\} \). They do not include fractions, decimals, or negative numbers.Understanding whole numbers is crucial when you are required to round estimates, like square roots, to the nearest whole number. For example, considering the estimate \(\sqrt{17.3} \approx 4\), we round down to the nearest whole number because 17.3 is closer to 16 than to 25. In applied scenarios, always consider how close the number is to neighboring whole numbers to decide whether to round up or down. This practice ensures you provide the best possible approximation without the need for complex calculation or computation tools.