Problem 51 - Elementary and Intermediate Algebra

Question

Use the power rule for exponents to simplify each expression. Write the results using exponents. $$ \left(3^{2}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(3^8\).

Step 1: Identify the Power Rule

The power rule for exponents states that when you raise a power to another power, you multiply the exponents. The rule is expressed as \((a^m)^n = a^{m imes n}\).

Step 2: Apply the Power Rule to the Expression

In the problem \((3^2)^4\), we identify \(3\) as the base, \(2\) as the first exponent \(m\), and \(4\) as the second exponent \(n\). Using the power rule, multiply the exponents: \(3^{2 \times 4}\).

Step 3: Simplify the Exponents

Calculate the exponent multiplication: \(2 \times 4 = 8\). Therefore, the expression simplifies to \(3^8\).

Step 4: Write the Final Answer

The simplified expression, using exponents, is \(3^8\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power or exponent. This exponent tells you how many times to multiply the base by itself. For example, in the expression

• \(3^2\), the base is 3, and the exponent is 2
• This means you multiply 3 by itself 2 times, resulting in \(3 \times 3 = 9\)

Exponentiation is a powerful tool in mathematics because it allows for concise expression of very large numbers, especially when dealing with repeated multiplication. It's an essential concept not only in algebra but also in many fields such as physics, computer science, and finance. When you raise an exponent to another power, as in the original exercise \((3^2)^4\), a rule called the "power rule" helps simplify the process.

Simplifying Expressions

Simplifying expressions refers to the process of transforming them into their simplest, most compact form, while maintaining their original value. This often involves processes such as combining like terms, factoring, reducing fractions, or applying known algebraic rules, like the power rule.
For example, with an expression like \((3^2)^4\), simplifying involves applying the power rule to reduce the expression to a single, simpler form:

• Identify the structure: Recognize the expression needs simplification.
• Apply the power rule: Multiply the exponents to simplify.
• Result: The simplified expression becomes \(3^8\).

These steps make the expression easier to understand and work with, especially in more complex mathematical problems. Simplification is crucial for efficient algebraic manipulation and solving equations.

Algebraic Expressions

Algebraic expressions are mathematical phrases containing numbers, variables, and operations. They are used to represent quantities and relationships in algebra. A typical algebraic expression can be something like \(x^2 + 3x + 2\) or the simpler \(3^2\), featuring

• Constants: Numbers on their own, like 2 or 3.
• Variables: Symbols, like \(x\), that represent unknown values or numbers that can vary.
• Operations: Such as addition, subtraction, multiplication, and division.

In the given exercise \((3^2)^4\), the expression involves exponentiation, which is a more advanced operation seen within algebraic expressions. Understanding how to manipulate and simplify these structures is the foundation of algebra, enabling you to solve equations and understand mathematical relationships in a variety of contexts. Algebraic expressions help translate real-world problems into mathematical form, making them solvable through logical deductions.