Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Approximate each expression to the nearest hundredth. $$\sqrt{[-1-(-3)]^{2}+(-5-3)^{2}}$$

Step-by-Step Solution

Answer

The expression approximates to 8.25.

Step 1: Simplify Inside Brackets

We start with the expression \(\sqrt{[-1-(-3)]^{2}+(-5-3)^{2}}\). First, resolve the terms inside the brackets.For \([-1-(-3)]\), the expression becomes \([-1+3]=2\).For \([-5-3]\), the expression becomes \([-5-3]=-8\).

Step 2: Compute Squares

Next, compute the squares of the results from Step 1.The square of 2 is \(2^2 = 4\), and the square of -8 is \((-8)^2 = 64\).

Step 3: Sum the Squares

Add the squared values from Step 2 together.We have \(4 + 64 = 68\).

Step 4: Calculate the Square Root

Find the square root of the sum obtained in Step 3.The square root of 68 is \(\sqrt{68} \approx 8.2462\).

Step 5: Round to the Nearest Hundredth

Round the result from Step 4 to the nearest hundredth.\(8.2462\) becomes \(8.25 \) when rounded to the nearest hundredth.

Key Concepts

Square Root Calculation

Square root calculation is an essential skill in mathematics that helps to determine the number which, when multiplied by itself, gives the original number. In the step-by-step solution you encountered, the square root calculation is carried out in Step 4 for the expression \( \sqrt{68} \). Square roots are often not whole numbers, which means that you may need to approximate them.
If you're working with an exact square root, such as \( \sqrt{64} = 8 \), the operation is straightforward. However, many square roots are not perfect squares, requiring some computation, such as using a calculator or estimation methods.

• To approximate \( \sqrt{68} \), you realize this number lies between perfect squares \( 64 \) and \( 81 \).
• Using a calculator or estimation, you find that \( \sqrt{68} \approx 8.2462 \).

Mastering square root calculations is crucial when working with distances and other geometric problems.

Rounding Numbers

Rounding numbers is a method used to simplify a number while maintaining its value close to the original number. It is particularly useful when dealing with long decimals. In solving our expression, the last step required rounding \( 8.2462 \) to the nearest hundredth, giving \( 8.25 \).
Here's how you can approach rounding:

• Identify the place to which you're rounding. For example, the hundredth place is two digits after the decimal point.
• Look at the digit immediately to the right of your target place. If it is 5 or more, round up. If it's less than 5, round down.

Using these rules, since the digit after the hundredth place in \( 8.2462 \) is 4, you round down, resulting in \( 8.25 \). Rounding simplifies numbers for easier communication and calculation.

Expression Simplification

Expression simplification is a foundational process in algebra where you reduce an expression to its simplest form without changing its value. This often involves performing operations like addition, subtraction, and division within the expression. In the context of our exercise, simplification was key in reducing \( [-1 - (-3)]^2 + (-5 - 3)^2 \) into a simpler form.
Here's how simplification was carried out step by step:

• First, simplify inside the brackets: convert \( -1 - (-3) \) to \( 2 \) and \( -5 - 3 \) to \( -8 \).
• Next, compute the squares of these simplified terms, resulting in \( 2^2 = 4 \) and \( (-8)^2 = 64 \).
• Add the squares together: \( 4 + 64 = 68 \).

These steps demonstrate how breaking down an expression into its basic parts can make complex arithmetic manageable. Understanding how to effectively simplify expressions is critical for tackling more advanced mathematics.