Problem 32 - Prealgebra

Question

Compute the exact square root. If the square root is undefined, write "undefined". \(\sqrt{100}\)

Step-by-Step Solution

Answer

The square root of 100 is 10.

Step 1: Identify the Problem

We are tasked with finding the exact square root of the number 100. The expression given is \( \sqrt{100} \).

Step 2: Understand Square Roots

The square root of a number \( x \) is a value \( y \) such that \( y^2 = x \). In simpler terms, it asks us which number, when multiplied by itself, results in 100.

Step 3: Calculate the Square Root

We need to find a number \( y \) such that \( y^2 = 100 \). We know that \( 10 \times 10 = 100 \), therefore \( 10^2 = 100 \).

Step 4: Conclusion Based on Calculations

Since \( 10 \times 10 = 100 \), the square root of 100 is \( 10 \). There is no ambiguity, as 100 is a perfect square.

Key Concepts

Perfect Squares

Perfect squares are numbers that result from squaring a whole number. In simpler terms, if you take a whole number and multiply it by itself, you get a perfect square.

For example:

• 3 squared is 9, written as:
  \(3^2 = 9\). Here, 9 is a perfect square.
• Similarly, 5 squared is 25, or:
  \(5^2 = 25\). This makes 25 a perfect square.

Understanding perfect squares helps us find square roots easily. If a number is a perfect square, its square root will be a whole number. In the case of 100, we know it's a perfect square because 100 results from \(10 \times 10\).

This characteristic of perfect squares makes solving square root problems more straightforward, especially in exams and tests.

Mathematical Operations

When working with mathematical operations, it's essential to understand the basic functions like addition, subtraction, multiplication, and division. Here, we deal with multiplication mainly in the context of squaring numbers.

To check whether a number is a perfect square, you perform a multiplication operation. For example, with 100, we multiply 10 by itself: \(10 \times 10 = 100\).

• Multiplication helps confirm whether a number can be squared back to a perfect square.
• Division can sometimes help in determining square roots, as dividing the number repeatedly can help identify factors.

Understanding these basic mathematical operations allows us to find solutions logically and efficiently. Mathematical operations are the foundation of more complex calculations and problem solving.

Prealgebra Concepts

Prealgebra serves as the stepping stone into the world of algebra. It introduces concepts that are pivotal for solving equations, including square roots.

Square roots are often introduced in prealgebra as part of a more extensive study on powers and exponents. Here, students learn that:

• Every number has a root, and finding that root means identifying what number can be raised to an exponent to obtain the original number.
• Understanding square roots requires recognizing numbers that are perfect squares.

In prealgebra, students also learn the properties of numbers and how those relate. Recognizing which numbers can be routinely squared or rooted, like 100, simplifies many future mathematical challenges.

By mastering prealgebra concepts, learners become proficient in solving not only square root problems but a variety of numerical tasks efficiently.