Problem 56 - Prealgebra

Question

Compute the exact value of the given expression. \(\sqrt{7^{2}+24^{2}}\)

Step-by-Step Solution

Answer

The exact value of the expression is 25.

Step 1: Simplify the expression inside the square root

First, calculate the squares of the two numbers inside the square root: 7 and 24. Calculate: 7 squared is: \[ 7^2 = 49 \]24 squared is: \[ 24^2 = 576 \]Sum these two results:\[ 49 + 576 = 625 \]

Step 2: Take the square root of the sum

Now, calculate the square root of the sum we obtained:The square root of 625 is:\[ \sqrt{625} = 25 \]

Key Concepts

Squaring Numbers

Squaring a number is simply multiplying the number by itself. It is one of the foundational operations in mathematics and is represented by raising a number to the power of 2, such as in the expression \( a^2 \). Let's break it down further to make it crystal clear.

• When you square the number 7, you perform the operation \( 7 \times 7 \), which equals 49.
• Similarly, squaring 24 is calculating \( 24 \times 24 \), resulting in 576.

Understanding squaring is essential as it comes up often in algebra and geometry problems. Squaring numbers can easily be done with a calculator, but knowing how to do it mentally or with pen and paper could be beneficial for quick checks and mental math.

Addition of Squares

Once you have squared each number, you move to adding these squares together. This is a straightforward arithmetic step but crucial for operations involving multiple squared terms. In our example:

• We squared 7 to get 49.
• We squared 24 to obtain 576.
• Add them together: \(49 + 576 = 625\).

This step is important in many areas of mathematics, especially when dealing with the Pythagorean Theorem or any distance formulas in geometry where the sum of squares frequently appears. By understanding the addition of squares, you will have an easier time navigating these geometric concepts.

Exact Values in Geometry

In geometry, exact values can often be crucial when solving problems, especially when calculating distances or when precise measurements are required. The expression \( \sqrt{7^2 + 24^2} \) is reminiscent of a Pythagorean triple (such as 3, 4, 5) which is a fundamental element in geometry.

After squaring and adding the numbers, you calculate the square root, in this case, \( \sqrt{625} \), to find the exact distance or length, resulting in 25. This reflects a relationship seen in right triangle sides, where the sum of the squares of the two shorter sides equals the square of the hypotenuse.

• Exact values are essential in drafting plans or creating models where precision is non-negotiable.
• It's also useful in theoretical problems, where an exact value conveys more meaning than an approximate decimal.

Mastering the computation of these values will aid you in both practical settings and advanced theoretical work in mathematics.