Problem 72 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The approximation is 8.25.

Step 1: Statement of the Problem

We need to approximate the expression \( \sqrt{[-1-(-3)]^2 + (-5-3)^2} \) to the nearest hundredth.

Step 1: Simplify Inside the Brackets

Start by simplifying the expressions inside the brackets: \[ [-1 - (-3)] = [-1 + 3] = 2 \]\[ [-5 - 3] = [-5 - 3] = -8 \] This expression simplifies to: \[ \sqrt{[2]^2 + (-8)^2} \]

Step 2: Calculate the Squares

Calculate the squares of both terms:\[ (2)^2 = 4 \] \[ (-8)^2 = 64 \] Thus, we have \[ \sqrt{4 + 64} \]

Step 3: Add the Squares

Add the squares obtained in the previous step:\[ 4 + 64 = 68 \] This gives us \[ \sqrt{68} \]

Step 4: Calculate the Square Root

Find the square root of 68:\[ \sqrt{68} \approx 8.246 \] Now, round this value to the nearest hundredth: \[ 8.25 \]

Key Concepts

Square Root Approximation

Approximating square roots is an important skill in mathematics, often leading to a more practical understanding of calculations without a calculator. To approximate the square root of a number like 68, we aim to find a whole number or a decimal that gets close to the true value of the root.

Here's how it's done:

• First, identify the perfect squares nearest to the number in question. For 68, this would be 64 (since \(\sqrt{64} = 8\)) and 81 (since \(\sqrt{81} = 9\)). Therefore, \(\sqrt{68}\) should fall between 8 and 9.
• Knowing this, you'd estimate the more precise value by testing midpoints (e.g., 8.2, 8.3, etc.) until you've approximated to your desired precision.
• Once you identify \(\sqrt{68} \approx 8.246\), rounding to the nearest hundredth gives us 8.25.

Comparing with known squares helps inform whether to adjust the estimate up or down. This process emphasizes estimation and provides intuition on how "close" a number is to being a perfect square.

Simplifying Expressions

Simplifying expressions is a fundamental step in solving many math problems. In our original exercise, the expression \([-1 - (-3)]\) is encountered, which can be a bit tricky at first glance.

Here's how simplification works in this context:

• Begin by eliminating double negatives. For example, \(-(-3)\) becomes \(+3\).
• So, \([-1 - (-3)] = [-1 + 3] = 2\).
• Similarly, for \([-5 - 3]\), you simply calculate it as usual, obtaining \(-8\).

Breaking down expressions into simpler parts not only makes them easier to work with but also reduces chances of making errors in your computations. In this way, working through each part incrementally is often the key to seeing the bigger picture of an equation.

Calculating Squares

Calculating squares means multiplying a number by itself. This concept is key when using the Pythagorean Theorem or other geometric and algebraic formulas.

To find the square of a number:

• Multiply the number by itself. For example, \(2^2\) becomes \(2 \times 2 = 4\), and \((-8)^2\) becomes \((-8) \times (-8) = 64\).
• Ensure careful handling of negative numbers. The square of a negative number is always positive, since two negatives make a positive.

In our problem, these calculations result in \(4\) and \(64\) respectively. Adding these results gives \(68\), which is then used for the square root calculation. Understanding squares is a foundation that supports many areas in algebra and geometry, emphasizing their importance in overall mathematical literacy.