Problem 55 - Algebra 2

Question

Simplify each radical expression. \(\sqrt{81}\)

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{81} \) is 9.

Step 1: Identify the Expression

First, examine the given expression, which is the square root of 81: \( \sqrt{81} \). This expression requires finding a number that, when squared, equals 81.

Step 2: Find the Square Root

Determine the square root of 81 by identifying a number whose square is 81. Since \( 9 \times 9 = 81 \), we know that the square root of 81 is 9.

Step 3: Express the Simplified Result

Having identified that 9 is the number such that when squared equals 81, the square root is simplified to: \( \sqrt{81} = 9 \). Therefore, the fully simplified form of the expression is 9.

Key Concepts

Simplifying Radicals

Simplifying radicals is a process that makes radical expressions more understandable and manageable. Radicals often involve square roots, cube roots, or other roots. The goal is to find a simpler or more precise way to express them.

To simplify a radical:

• Identify the radicand, which is the number inside the radical sign.
• Look for factors of the radicand that are perfect squares (such as 4, 9, 16, etc.). For cube roots, look for perfect cubes, and so on.
• Rewrite the radical expression using these factors.
• Simplify by taking the square root of the perfect square factor, leaving anything else under the radical.

Take \( \sqrt{81} \). Identifying 81 as a perfect square makes it straightforward because \( 81 = 9 \times 9 \). This means \( \sqrt{81} \) simplifies directly to 9.

Square Roots

A square root is a value that can be multiplied by itself to give the original number. When you see \( \sqrt{} \), it refers to the principal square root, which is the non-negative root. For example, 9 is the square root of 81 because \( 9 \times 9 = 81 \).

Key points about square roots:

• The square root of a positive number is always non-negative.
• Perfect squares have exact whole number roots: \( \sqrt{16} = 4 \), \( \sqrt{25} = 5 \).
• If a number isn't a perfect square, its square root will be irrational, such as \( \sqrt{2} \) or \( \sqrt{3} \).
Learn more about square roots here.

The process of square rooting involves finding that specific number which, when squared, results in the radicand. Simplifying radicals involves this concept.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (like +, −, ×, ÷) that represent a value. Radical expressions are a type of mathematical expression that includes a root symbol, like a square root.

When working with mathematical expressions involving radicals:

• Understand each component, such as constants or variables.
• Look for opportunities to simplify by factoring out perfect squares or cubes.
• Combine like terms for expressions that may include similar radical terms.
• Always aim for an expression that is easy to interpret and as simplified as possible.

Simplifying a radical expression involves these general principles of simplifying any mathematical expression—building a deep understanding of each part to make the entire expression clearer.