Problem 55 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$ \sqrt{13} $$

Step-by-Step Solution

Answer

The approximate value of \( \sqrt{13} \) to the nearest hundredth is 3.61.

Step 1: Identify the number under the square root

In this case, the number under the square root or radicand is 13.

Step 2: Evaluate the square root

As 13 is not a perfect square, its square root won't be an exact whole number. Therefore, use a calculator or a similar tool to calculate the square root.

Step 3: Approximate to the nearest hundredth

After finding the square root of 13 from step 2, round this number to the nearest hundredth place. That is the second digit after the decimal point.

Key Concepts

Square Root Approximation

When we come across a square root that isn't a perfect square, we need to find an approximate value. This approximation can be derived either by using estimation techniques or with a calculator. To manually approximate a square root, we can use the method of finding two perfect squares that the radicand falls between. For instance, to approximate \( \sqrt{13} \), we know that \( 13 \) lies between the perfect squares of \( 3^2 = 9 \) and \( 4^2 = 16 \) because \( 3 < \sqrt{13} < 4 \). With this information, we can deduce that \( \sqrt{13} \) is closest to \( 3.xxx \). To improve the accuracy of this exercise, we can then apply a more numerical method, such as finding the decimal part with a calculator.

Using a calculator to find the square root gives us a more precise decimal number before rounding. However, understanding the concept of approximation, especially when calculators are not permitted, is crucial. Knowing which perfect squares your radicand is between sets a strong foundation for estimating square roots.

Radicand

The radicand is the number under the square root symbol, indicated by \( \sqrt{\text{radicand}} \). It's the 'subject' of your square root operation. In our example, \( 13 \) is the radicand inside \( \sqrt{13} \). The value of the radicand greatly determines the approach to evaluating the square root. If the radicand is a perfect square—like \( 9 \) or \( 16 \)—the square root is a whole number. However, if it's not a perfect square—as with \( 13 \)—we have to estimate its square root or use a calculator to find a decimal approximation.

Understanding the radicand's role is essential since it affects how we approach the problem. If we inadvertently alter the radicand, the entire meaning and therefore the result of the square root change dramatically. It's important to keep the radicand intact throughout the calculation process.

Rounding Decimals

After calculating an approximation of a square root, we often need to round the result to a specified level of accuracy. Rounding decimals to the nearest hundredth means looking at the third decimal place. If it is five or higher, you increase the second decimal place by one. Otherwise, you leave it as is. For example, if a square root approximation calculator yields a value of \( 3.60555 \) for \( \sqrt{13} \), we would round this to \( 3.61 \).

This process is not just about chopping off numbers after a certain decimal place; it's about accurately representing numbers in a simplified form while maintaining the value's integrity as closely as possible. Whether you're dealing with money, measurements, or statistics, learning to round decimals correctly is crucial. In the context of square roots, it allows us to communicate and record our findings with clarity and precision.