Problem 44 - Elementary and Intermediate Algebra

Question

Square or cube each quantity and simplify the result. $$ (\sqrt{11})^{2} $$

Step-by-Step Solution

Answer

The result is 11.

Step 1: Understand the Problem

We are asked to square the expression \( \sqrt{11} \), which means raising it to the power of 2.

Step 2: Apply the Squaring Operation

When you square a square root, you essentially eliminate the square root. Mathematically, this is because \( (\sqrt{a})^2 = a \). In this case, \( (\sqrt{11})^2 = 11 \).

Step 3: Simplify the Expression

The expression \( (\sqrt{11})^2 \) simplifies directly to \( 11 \). There are no further steps needed since \( 11 \) is already in its simplest form.

Key Concepts

Square Root

A square root of a number is a value that, when multiplied by itself, gives the original number. The square root is denoted by the radical symbol \( \sqrt{} \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). The process of taking a square root is essentially the reverse of squaring. When you square a square root, the operations undo each other, resulting in the original number.
- Example: \( \sqrt{11} \) is the number that when multiplied by itself results in 11.
- If we have \( (\sqrt{11})^2 \), squaring this expression nullifies the square root, giving us simply \( 11 \).
This principle is true for all numbers and is a foundational concept in understanding how different mathematical operations interact. Remember, the square root only applies to non-negative numbers to yield real number results.

Mathematical Simplification

Simplification in mathematics involves reducing a complex expression into a simpler or more manageable form. This is often done by eliminating unnecessary components or simplifying operations.
For square roots, this can mean removing the radical sign if it's within a power, like in our example where \( (\sqrt{11})^2 = 11 \). Simplification helps in:

• Making expressions easier to understand
• Facilitating further operations
• Reducing errors in calculations

An expression is considered fully simplified when no further reduction is possible, as with \( 11 \) in our exercise. Proper simplification often depends on recognizing and applying mathematical rules effectively, such as pairing operations that cancel each other out.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations like addition, subtraction, multiplication, and division. In our exercise, \( \sqrt{11} \) is a simple algebraic expression involving a square root.
Key elements include:

• Constants: Fixed numerical values like \( 11 \)
• Variables: Symbols that represent numbers
• Operators: Symbols denoting mathematical operations

The goal in working with algebraic expressions is often to manipulate them by applying various operations and rules, to either simplify or solve for specific values. It's this fundamental capability in algebra that allows for solutions to complex problems in mathematics, science, and engineering. Understanding these expressions requires practice in manipulating numbers and understanding how different terms and operations interact.