Problem 54 - Algebra 2

Question

Simplify each radical expression. \(\sqrt{25}\)

Step-by-Step Solution

Answer

The simplified expression is 5.

Step 1: Identify the Radicand

The given radical expression is \(\sqrt{25}\). The number 25 inside the radical symbol is called the radicand.

Step 2: Determine the Square Root

We need to find the square root of 25. Think of a number which, when multiplied by itself, gives 25.

Step 3: Calculate the Square Root

The number is 5, because \(5 \times 5 = 25\). Therefore, \(\sqrt{25} = 5\).

Step 4: Write the Simplified Expression

The simplified form of the given radical expression \(\sqrt{25}\) is 5.

Key Concepts

Square Root

The square root is a fundamental concept in mathematics. It represents finding a number that produces a specified value when multiplied by itself. For example, when we see the expression \(\sqrt{25}\), we're tasked with identifying the square root of 25.

Understanding square roots is crucial because it forms the basis of many advanced mathematical concepts. The square root of a number \(n\) is a value that, when squared (multiplied by itself), gives \(n\). Hence, the expression \(\sqrt{n}\) implies searching for that specific value.

For example:

• The square root of 4 is 2, because \(2 \times 2 = 4\).
• The square root of 9 is 3, as \(3 \times 3 = 9\).

Finding square roots can become routine with practice. It involves recognizing perfect squares, which are numbers like 1, 4, 9, 16, and so forth, that result from squaring whole numbers.

Radicand

The term radicand refers to the number which is placed beneath the radical sign, or the square root symbol \(\sqrt{}\).

In the context of the expression \(\sqrt{25}\), the number 25 acts as the radicand. This is the value of which we are \(trying\) to find the root. Understanding the concept of a radicand is important for simplifying expressions and performing mathematical operations on radicals.

To work effectively with radicals, it's crucial to identify the radicand with ease. This helps streamline the process of solving and simplifying radical expressions. Different radicands lead to different results, so recognizing them accurately is vital when attempting to manipulate or simplify these expressions.

Once you have the radicand, you can proceed to calculate its square root, simplifying the expression further.

Mathematical Operations

Mathematical operations are the actions performed to reach a solution or simplify an expression. In the context of radical expressions, these operations are key to understanding how to break down more complex problems into simpler parts.

For example, when given \(\sqrt{25}\), you calculate using the operation of finding a square root, which is fundamental in simplifying radicals. The operation requires finding a number that multiplies by itself to produce the radicand.

Key operations related to radicals include:

• Addition and Subtraction: Like with polynomials, combine only like terms. Radicals must have the same radicand and index to be added or subtracted.
• Multiplication and Division: When multiplying or dividing radicands, perform the operation as you'd normally, and adjust the radical form afterward if needed.

Understanding and applying these operations help make solving radical expressions straightforward and manageable, often reducing them to simpler, more recognizable forms.