Problem 62 - Prealgebra

Question

Compute the exact value of the given expression. \(\sqrt{6^{2}+8^{2}}\)

Step-by-Step Solution

Answer

The exact value is 10.

Step 1: Understand the Expression

The expression given is \(\sqrt{6^{2}+8^{2}}\). This is asking us to find the square root of the sum of the squares of the numbers 6 and 8.

Step 2: Calculate the Individual Squares

First, calculate the square of 6: \[ 6^2 = 36 \]Next, calculate the square of 8: \[ 8^2 = 64 \]

Step 3: Add the Squares Together

Add the results of the squares from the previous step:\[ 36 + 64 = 100 \]

Step 4: Find the Square Root

Now, find the square root of the sum calculated in Step 3:\[ \sqrt{100} = 10 \]

Step 5: Finalize the Answer

The exact value of the expression \(\sqrt{6^2 + 8^2}\) is 10.

Key Concepts

Calculating Square Roots

The square root operation helps us find the original number that was multiplied by itself to get a certain value. In the expression \(\sqrt{6^2 + 8^2}\), we ultimately want to find the square root of the sum of 36 and 64, which equals 100. Calculating square roots can reveal prime numbers or simplify complex expressions. It's like working backwards from the square number to its root.
For instance:

• The square root of 4 is 2, since \(2 \times 2 = 4\).
• Similarly, the square root of 100 is 10, because \(10 \times 10 = 100\).

To calculate a square root, think of it in terms of division or using a calculator to quickly find the answer. Understanding square roots simplifies solving problems quickly, just as we did above to achieve the result of 10 for our expression.

The Addition of Squares

The addition of squares is a crucial concept when solving equations involving the Pythagorean Theorem. In the expression \(\sqrt{6^2 + 8^2}\), adding squares means we compute each number raised to the power of two first, then add them together.
Here’s what it looks like:

• Calculate \(6^2\) to get 36.
• Calculate \(8^2\) to get 64.
• Add them together: \(36 + 64 = 100\).

This process helps us in many areas of math, beyond just solving right triangles.
When adding squares, ensure calculations are accurate by double-checking each square. This verifies the final result is correct before you proceed to the next step, like finding a square root.

Right Triangle Computation

Right triangle computation is made considerably easier with the Pythagorean Theorem. This theorem states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the longest side. When we looked at \(\sqrt{6^2 + 8^2}\), we applied this principle.
Imagine a triangle where one side is 6 units, and another side is 8 units:

• The calculation for the hypotenuse, or the longest side, becomes trivial using this theorem: \(36 + 64 = 100\).
• The square root of 100 tells us the hypotenuse is 10 units long.

Understanding this process allows you to solve similar problems, identify unknown sides of right triangles, or verify that a triangle is indeed right-angled. The simplicity of these computations means they can be used in geometric designs, navigation, and even real-world tasks requiring precise measurements involving triangles.