Problem 8 - Prealgebra

Question

Find each of the following square roots without using a calculator. $$\sqrt{169}$$

Step-by-Step Solution

Answer

The square root of 169 is 13.

Step 1: Understand the Square Root

The square root of a number, say \( n \), is a value that, when multiplied by itself, equals \( n \). For example, for \(  6 \), we want to find a number \( x \) such that \( x imes x = 169 \).

Step 2: Recall Perfect Squares

Recall that some numbers are perfect squares, which means they are the product of an integer multiplied by itself. For instance, \( 10 \times 10 = 100 \), \( 11 \times 11 = 121 \), \( 12 \times 12 = 144 \), and \( 13 \times 13 = 169 \).

Step 3: Identify the Square Root

Identify that since \( 13 imes 13 = 169 \), the square root of \( 169 \) must be the number \( 13 \). Thus, \(   13 = 169 \).

Key Concepts

Understanding Perfect Squares

Perfect squares are numbers that result from multiplying an integer by itself. They are fundamental when dealing with square roots.
Recognizing perfect squares can simplify many mathematical problems, especially those that involve finding square roots without a calculator.

• Examples of perfect squares include 1, 4, 9, 16, 25, 36, and so on.
• These come from \(1 \times 1, 2 \times 2, 3 \times 3, 4 \times 4, 5 \times 5, 6 \times 6,\) etc.
• Perfect squares grow more distant from each other as the numbers increase, such as \(144 = 12 \times 12\) and \(169 = 13 \times 13\).

When you know perfect squares, finding square roots becomes simply a matter of knowing these specific products. If you spot that a number is a perfect square, you instantly know its square root.

Basics of Multiplication

Multiplication is one of the fundamental arithmetic operations. When we multiply two numbers, we are essentially adding one number to itself a certain number of times. Understanding this operation is key for grasping how square roots and perfect squares work.

• For example, if we multiply 2 by 3, it is the same as adding 2 three times: \(2 + 2 + 2 = 6\), hence, \(2 \times 3 = 6\).
• With perfect squares, multiplication involves the same number repeated: \(13 \times 13 = 169\), simplifying to a more streamlined form: \(13^2\).
• Through multiplication, we can confirm if a number is a perfect square, as seen in \(13 \times 13 = 169\).

Thus, mastering multiplication is not just about calculating results, but recognizing patterns such as perfect squares, which has large implications in finding square roots.

Understanding Integers

Integers include all whole numbers and their negatives. They are crucial when discussing perfect squares and square roots.
In the context of square roots, we focus on non-negative integers since they lead to perfect squares.

• Examples of integers are \(-3, 0, 7, 13,\) and \(25\).
• When we talk about perfect squares like \(169,\) they are indeed integers because they result from \(13 \times 13\).
• Finding a square root means identifying a non-negative integer that when squared, matches the perfect square, like finding \(13\) for \(169\).

Thus, understanding integers helps in identifying and working with perfect squares and their corresponding square roots effectively.