Problem 23 - Intermediate Algebra

Question

Evaluate each square root without using a calculator. See Objective 1 and Example 1. $$ \sqrt{100} $$

Step-by-Step Solution

Answer

The square root of 100 is 10.

Step 1: Identify the Perfect Square

The given expression is \( \sqrt{100} \). First, recognize that 100 is a perfect square number because it can be expressed as an integer squared.

Step 2: Determine the Integer Root

Recall the definition of a square root: it is a number which when multiplied by itself yields the original square number. Since \( 10 \times 10 = 100 \), the square root of 100 is 10.

Step 3: Write the Solution

Having identified that 10 squared equals 100, you can now conclude that \( \sqrt{100} = 10 \).

Key Concepts

Perfect Square

A perfect square is a special type of number. It results when an integer is multiplied by itself. These numbers are important in mathematics because they are easy to work with when finding square roots. For example, the number 100 in the expression \( \sqrt{100} \) is a perfect square. This is because you can express 100 as \( 10 \times 10 \). Recognizing perfect squares can simplify complex problems. Some examples of perfect squares are:

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

Notice that each of these numbers can be expressed as the square of an integer. This is what makes them perfect squares. Identifying them will make finding square roots much easier.

Integer Root

In mathematical terms, an integer root is the solution of an equation that involves roots, and it is specifically an integer. When dealing with square roots, the integer root is the integer that when squared results in the original perfect square. Consider \( \sqrt{100} \) as an example. We are looking for an integer that squares to give 100. In this situation, the integer root is 10 because \( 10 \times 10 = 100 \). Identifying integer roots involves recognizing perfect squares and knowing the simplest multiplication facts. Here are some simple tips:

• Check if the number can be expressed as \( n \times n \) for an integer \( n \).
• Learn and memorize squares of integers to quickly find integer roots.
• Use the properties of perfect squares to recognize integer roots faster.

Knowing how to find integer roots can help solve equations and simplify expressions.

Mathematical Expression

A mathematical expression combines numbers, operators, and sometimes variables to convey a specific value or relation. Expressions can be as simple as a single number or complex with multiple operations. We can see \( \sqrt{100} \) as a mathematical expression representing a square root operation. Expressions involving square roots often aim to simplify or evaluate them. Here’s a quick way to approach these types of expressions:

• Identify if the number underneath the square root is a perfect square. This simplifies the expression significantly.
• Apply the properties of square roots, such as \( \sqrt{x^2} = x \), for positive real numbers \( x \).
• Simplify expressions step by step by breaking down calculations into smaller, more manageable parts.

Understanding and working with mathematical expressions is crucial in not just finding square roots but all areas of math. It forms the basis for calculations, problem-solving, and mathematical reasoning.