Problem 10 - College Algebra with Modeling and Visualization

Question

Find the principal square root of the number. Approximate your answer to the nearest hundredth whenever appropriate. $$ 45 $$

Step-by-Step Solution

Answer

The principal square root of 45 is approximately 6.71.

Step 1: Understanding Principal Square Root

The principal square root of a number is the non-negative root. Our task is to find the square root of 45.

Step 2: Use a Calculator

To find the principal square root of 45, use a calculator. The principal square root of 45 is approximately: \[\sqrt{45} \approx 6.708\]

Step 3: Round to Nearest Hundredth

Round the calculated square root (approximately 6.708) to the nearest hundredth. Because the third decimal is an 8, round up the second decimal to get:\[6.71\]

Key Concepts

Principal Square Root

The principal square root of a number is simply the non-negative square root. When we talk about the square root, we might think of both positive and negative roots because squaring either of these values will result in the original number. For instance, both 3 and -3 are square roots of 9. However, the principal square root is only the positive one, which is 3 in this case.

For the number 45, the task is to find the non-negative root. This just means finding the positive number that, when multiplied by itself, results in 45. It’s helpful to remember that by convention, when we talk about squares and square roots, the principal square root is the go-to unless stated otherwise.

• Always choose the non-negative root for principal square root problems.
• It's important especially when dealing with real numbers in real-world applications.

Approximation

Approximating square roots is crucial because many square roots are irrational numbers, meaning they have infinite non-repeating decimals. In our example, the square root of 45 is not a simple whole number, but rather a decimal that continues indefinitely.

In mathematics and practical applications, it's common to approximate such values to make calculations manageable and results more understandable. In this case, using a calculator gives us the result as \( \sqrt{45} \approx 6.708 \).

Approximation allows us to make sense of complex numbers, especially in fields like science and engineering where precision is important but absolute values are impractical. Always remember:

• Approximation simplifies complex numbers and makes them usable.
• It helps in obtaining a practical and usable answer.

Rounding Numbers

Rounding numbers is a method we use to simplify numbers, by trimming them to a number of decimal places that make sense for the situation we're analyzing. In our step-by-step solution, we approximated \( \sqrt{45} \approx 6.708 \) and were then asked to round this to the nearest hundredth.

Here's how rounding to the nearest hundredth works: look at the third decimal place. If it's 5 or more, round the second decimal place up by one. Conversely, if it's less than 5, simply leave the second decimal place as it is. In our example of \( 6.708 \), the third decimal is "8", so we round up the "0" in the second decimal place to a "1". Thus, \( 6.71 \) is the rounded value.

Rounding numbers is a useful skill that:

• Simplifies figures for easy communication and comprehension.
• Helps achieve consistency in reporting and data handling.

By understanding and applying rounding, approximate results are made both practical and easy to interpret.