Problem 44 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The simplified result is 11.

Step 1: Understand the Problem

The problem is asking us to square a square root expression, specifically \( (\sqrt{11})^2 \). Our goal is to find the simplified form of this expression.

Step 2: Recall the Property of Exponents and Radicals

We know that squaring a square root removes the square root. Mathematically, this principle is expressed as \((\sqrt{a})^2 = a\). Hence, \((\sqrt{11})^2\) would equal 11.

Step 3: Apply the Property to the Given Expression

Based on the property \((\sqrt{a})^2 = a\), we apply it to \((\sqrt{11})^2\) to get \(11\). This is because the square root and the square are inverse operations.

Step 4: Simplify the Result

Since the expression simplifies directly to \(11\), our work is complete. No further simplification is necessary.

Key Concepts

Simplification of Expressions

Simplification is all about transforming complex expressions into their simplest form while maintaining their original value. In algebra, this process often involves reducing or condensing expressions by combining like terms and using known mathematical identities.

• Identification: First, identify any parts of the expression that can be combined or simplified, such as similar terms or mathematical operations that cancel each other out.
• Application of Rules: Use algebraic rules and properties. Simplifying expressions often requires the use of laws of exponents, such as understanding that squaring a square root returns the original number.
• Verification: Always double-check your work to confirm that the simplified expression is equivalent to the original.

In the case of \((\sqrt{11})^2\), once the property of exponents and radicals is applied, the expression simplifies immediately to 11, signifying the power of these simplification techniques.

Properties of Exponents

Exponents are mathematical notations indicating the number of times a number, known as the base, is multiplied by itself. Understanding their properties allows for the simplification of complex algebraic expressions efficiently.

• Multiplication Property: When multiplying two powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
• Division Property: When dividing two powers with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: Raising a power to another power means you multiply the exponents: \((a^m)^n = a^{m \times n}\).

In the original exercise, \((\sqrt{11})^2\) demonstrates the property \((\sqrt{a})^2 = a\), which shows how exponents and radicals interact as inverse operations. Understanding this property helps simplify expressions involving roots and exponents.

Algebraic Operations

Algebraic operations include basic arithmetic operations such as addition, subtraction, multiplication, and division, and extend to the handling of expressions involving exponents and radicals.

• Addition and Subtraction: Combine like terms, which have the same base and power, effectively.
• Multiplication and Division: Implement the multiplication and division properties of exponents to simplify expressions involving powers.
• Working with Radicals: Recognize that radicals, specifically square roots, can often be simplified or altered using the relationship between exponents and roots.

These operations require understanding the associated rules and how they interact within expressions. For instance, in the exercise where \((\sqrt{11})^2 = 11\), multiplication is utilized where a square root multiplied by itself removes the radical, simplifying it algebraically.