Question

Simplify each expression. \(\sqrt{100}\)

Short answer

The simplified form of \(\sqrt{100}\) is 10.

Step-by-step solution

Understand the Problem

The task is to simplify the square root of 100, which involves finding a number that, when multiplied by itself, gives 100.

Perfect Square Recognition

Recognize that 100 is a perfect square. This means there is a whole number whose square is 100. The number is 10 because \(10 \times 10 = 100\).

Apply the Square Root Property

For perfect squares, the square root simplifies to the whole number forming the square. The expression \(\sqrt{100}\) becomes simply 10.

Key concepts

Perfect Squares

In mathematics, a **perfect square** is a number that can be expressed as the square of an integer. In other words, if you multiply an integer by itself, the result is a perfect square. For example, numbers like 1, 4, 9, 16, 25, and so on are perfect squares because

• 1 is 1 × 1,
• 4 is 2 × 2,
• 9 is 3 × 3, and
• 16 is 4 × 4.

Recognizing perfect squares is essential when working with square roots. If the number under the square root (the radicand) is a perfect square, the square root simplifies directly to an integer.
For instance, in the problem offered, 100 is identified as a perfect square because it equals 10 × 10. This recognition allows us to state immediately that the square root of 100 is 10.

Simplification

**Simplification** is the process of reducing a mathematical expression to its simplest form. When simplifying expressions involving square roots, it's crucial to recognize if the radicand is a perfect square, as this can directly simplify into an integer.
If a number under the square root isn't a perfect square, it may not simplify perfectly to an integer, but sometimes, the expression can be reduced further by factoring out the perfect squares.

• For instance, \(\sqrt{50}\) can be simplified by recognizing that 50 is 25 × 2. Since 25 is a perfect square, \(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2}\).

In the given example of \(\sqrt{100}\), simplification is straightforward because 100 is a perfect square, so the simplified form of the expression is 10.

Mathematical Expressions

**Mathematical expressions** involve numbers, variables, and operations that represent a particular value or solve a problem. Square roots are pieces of mathematical expressions that specify the number of times a number must be multiplied by itself to produce a given number.
When dealing with expressions, simplifying them is often necessary to make understanding or further calculations easier and clearer. Consider the expression \(\sqrt{x}\). If \(x\) is replaced by a perfect square like 100, the expression becomes \(\sqrt{100}\), which simplifies to 10.
By simplifying mathematical expressions, we not only make problems easier to manage but also help ensure accuracy in mathematical problem-solving and reasoning. Understanding this concept and simplifying expressions, when needed, is a valuable skill and forms the foundation for more complex algebra and calculus work.