Problem 60 - Prealgebra

Question

Compute the exact value of the given expression. \(\sqrt{15^{2}+20^{2}}\)

Step-by-Step Solution

Answer

The exact value is 25.

Step 1: Square each number inside the square root

First, we need to square the numbers 15 and 20 separately. This means computing: \(15^{2} = 225\) and \(20^{2} = 400\).

Step 2: Add the squared values

Now, we add the squares of 15 and 20 together. So, compute: \(225 + 400 = 625\).

Step 3: Take the square root

Finally, take the square root of the sum from Step 2. The expression now is \(\sqrt{625}\), which simplifies to 25 because \(25 \times 25 = 625\).

Key Concepts

Square Root Calculation

Understanding square roots is crucial in solving problems like the Pythagorean theorem. A square root essentially "undoes" the squaring of a number. When you see an expression like \( \sqrt{625} \), you're looking for a number that, when multiplied by itself, equals 625. In our context, the square root of 625 is 25, because \( 25 \times 25 = 625 \).
Square roots are often related to solving for the length of a side in right-angled triangles, where the sides are calculated using squares and square roots.
Here’s a step-by-step thought process:

• Identify the number under the square root. In our case, that's 625.
• Think of perfect squares close to this number. Since 25 is a common perfect square, we use it to simplify.
• Verify by squaring the result. If \( 25 \times 25 \) equals 625, then your calculation is correct!

Exponents and Powers

Exponents indicate how many times you multiply a number by itself. In the expression \( 15^2 \) or \( 20^2 \), the 2 is the exponent, meaning we multiply 15 by itself once (\( 15 \times 15 \)) and similarly for 20. Calculating \( 15^2 = 225 \) means you get 225 after multiplying 15 two times. Similarly, \( 20^2 \) gives us 400.
Exponents make calculations efficient because instead of writing out multiple multiplications, you have a compact expression.
Some fascinating facts:

• \( x^1 \) is simply x, and \( x^0 \) is always 1, no matter what x is.
• The "square" of a number relates directly to areas in geometry.
• Learning some common squares, like \( 2^2 = 4 \) or \( 12^2 = 144 \), helps solve problems quickly.

Addition of Squares

The addition of squares is a foundational arithmetic operation, especially when working with the Pythagorean theorem. Here, you are adding two squared numbers to obtain another square whose root becomes the length of the hypotenuse in a triangle.
In our exercise, we added \( 15^2 \) and \( 20^2 \). That turns into adding \( 225 + 400 = 625 \). This step combines the squared sides into one expression which eventually becomes the square root calculation.Key insights:

• Addition of squares can often be used to verify measurements or solve for unknowns in geometry.
• It’s algebraically simple, meaning it doesn’t change the essence of individual squares.
• The result of this addition forms the 'c' part in the famous formula \( a^2 + b^2 = c^2 \) of right-angled triangles.