Problem 66 - Fundamentals of Mathematics

Question

Find the roots (using your knowledge of multiplication). Use a calculator to check each result. \(\sqrt{169}\)

Step-by-Step Solution

Answer

The square root of 169 is 13.

Step 1: Understand the Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we need to find a number that, when multiplied by itself, equals 169.

Step 2: Identify the Number

Consider the multiplication tables or known perfect squares to identify the number that squares to 169. Some common squares to remember are: \(10^2 = 100\), \(11^2 = 121\), \(12^2 = 144\), \(13^2 = 169\). Thus, 13 multiplied by itself is 169.

Step 3: Confirm the Root with a Calculator

Use a calculator to find the square root of 169 by pressing the square root button and entering 169. The calculator should return 13, confirming that 13 is the square root of 169.

Key Concepts

Perfect Squares

Perfect squares are numbers that are obtained by squaring whole numbers. This means multiplying a whole number by itself. For example, squaring the number 4 gives 16, which is a perfect square. Perfect squares are easy to remember, as they follow a predictable pattern. Here are a few examples:

• \(1^2 = 1\)
• \(2^2 = 4\)
• \(3^2 = 9\)
• \(4^2 = 16\)
• \(5^2 = 25\)

For the number 169, we identify it as a perfect square because it can be expressed as \(13 \times 13\). When learning about square roots, familiarizing yourself with the list of perfect squares can be incredibly helpful. This knowledge aids in quickly determining square roots for perfect squares. In this case, since 169 is a perfect square, we find that its square root is 13.

Multiplication

Multiplication is a fundamental arithmetic operation that quickly adds equal groups. When looking for the square root of a number, like 169, knowing how to manipulate multiplication tables is key. You look for a number that, when multiplied by itself, equals the original number. This process requires understanding the concept of perfect squares.
Here's how multiplication aids in finding square roots:

• You start with known perfect squares (e.g., \(10^2, 11^2, 12^2\), etc.).
• Compare them to your target (e.g., 169) to find the exact match. In this case, \(13^2 = 169\).
• Check by reverse multiplying (\(13 \times 13 = 169\)).

By breaking numbers into familiar multiplication patterns, you can quickly and accurately find square roots without always relying on technological devices. Multiplication is essential for this task as it allows checks and internal validation before using a calculator for confirmation.

Using a Calculator

Calculators are fantastic tools that simplify finding square roots, especially when dealing with large or unfamiliar numbers. However, for small and reasonably familiar numbers, such as 169, using prior mathematical knowledge and mental calculations is valuable.
Here's how to use a calculator effectively to find square roots:

• Press the button labeled with a square root symbol \(\sqrt{}\).
• Enter the number you need the square root for (e.g., 169).
• Press enter or the equals button to get your result.
• Depending on the calculator, it should display 13, confirming that 13 is the square root of 169.

Using a calculator is beneficial for confirming manual calculations or handling more complicated numbers. It's a good practice to verify the square root with multiplication as described above to enhance your understanding of the result. Combining mental math and calculators optimizes learning and problem-solving skills.