Problem 28 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$100$$

Step-by-Step Solution

Answer

The square roots of 100 are 10 and -10

Step 1: Identify the number and symbolize it

In this case, the number given is \(100\). The square root of \(100\) can be symbolized as \(\sqrt{100}\) or \(-\sqrt{100}\).

Step 2: Find the square roots

A square root of a number \(x\) is a number that, when multiplied by itself, gives the original number \(x\). To find the square roots of \(100\), we need to identify which number times itself equals \(100\). That would be \(10\) and \(-10\). Therefore the square roots of \(100\) are \(10\) and \(-10\).

Step 3: Verify the results

As a check, let's square each root. \(10^2 = 100\) and \((-10)^2 = 100\). Hence, both roots are correct.

Key Concepts

Number Theory and Square Roots

Number theory deals with the properties and relationships of numbers, particularly integers. When we talk about square roots in number theory, we refer to finding a number that, when multiplied by itself, gives a specified integer. For example, the number 100 is a perfect square because it can be expressed as the square of an integer, specifically 10 or -10.

This leads us to understand what a perfect square is—a number that has an integer as its square root. In the realm of number theory, determining whether a number is a perfect square involves checking if there exists an integer whose square equals that number. In our case, because 100 equals 10 squared or (-10) squared, it is classified as a perfect square.

• The task is to find such integers for any given perfect square.
• If integers exist, they are called the square roots of that number.

Connecting our understanding of number theory to practice, identifying square roots becomes a fundamental exercise in establishing relations between different sets of numbers.

Integer Mathematics and Perfect Squares

Integer mathematics is a branch of mathematics that concentrates on whole numbers, including both positive and negative, as well as zero. Square roots are incredibly relevant in integer mathematics because they often return integer results when starting with perfect squares.

For example, let's consider the number 100, which is a common example of a perfect square. The square roots of 100 are 10 and -10. Here

• The positive integer root, 10, and
• The negative integer root, -10,

demonstrate the symmetry found in integer mathematics when dealing with square roots.

This symmetry reflects a critical concept: each perfect square has two roots, namely a positive and a negative integer. This property is fundamental in integer mathematics as it showcases the balance and significance of integers in numerical operations.

Verification of Solutions

Verification is an essential part of solving mathematical problems to ensure that the solution is correct. In the context of square roots and integer solutions, verification involves squaring the possible roots to see if they return the original number.

Let's take square roots of 100, for instance: 10 and -10. To verify these roots, we square them:

• Squaring 10, we get 10², which equals 100.
• For -10, squaring gives us (-10)², which also equals 100.

By performing these operations, we confirm that our identified roots are correct. Verification thus acts as a crucial step in ensuring not only that our calculation is accurate but also that it makes logical sense.

This approach promotes confidence in a given solution, especially in mathematical problems that involve integer mathematics and number theory, where precision and correctness are paramount.