Problem 76 - Applied Calculus

Question

Simplify. $$ \left[\left(x^{3}\right)^{3}\right]^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^{27}\).

Step 1: Simplify the Inner Expressions Using Exponentiation Rule

The first step is to simplify the expression \( (x^3)^3 \) using the exponentiation rule \((a^m)^n = a^{m \cdot n}\).Apply the rule to the innermost parentheses:\[(x^3)^3 = x^{3 \cdot 3} = x^9\]

Step 2: Simplify the Outer Expression Using Exponentiation Rule Again

Now, replace \((x^3)^3\) with \(x^9\) in the original expression and simplify using the exponentiation rule again:\[(x^9)^3 = x^{9 \cdot 3} = x^{27}\]

Step 3: Combine Results for the Final Simplified Expression

After simplifying the inner and outer expressions, the entire expression becomes \(x^{27}\). This is the simplified form of the given expression.

Key Concepts

Exponentiation Rule

Exponentiation is a mathematical operation involving numbers called the base and power (or exponent). The exponentiation rule is a key concept, helping us efficiently work with powers. The rule can be expressed as \( (a^m)^n = a^{m \cdot n} \). This means when you raise a power to another power, you simply multiply the exponents. For example, in the expression \((x^3)^3\), we use this rule. By understanding and applying these rules correctly, you can simplify complex expressions neatly.
Applying the rule isn’t just about memorization. It's about seeing how numbers behave and relate. With practice, using the exponentiation rule becomes intuitive and straightforward.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form. This makes it easier to work with them in equations and calculations. When simplifying, you apply mathematical operations like addition, subtraction, multiplication, division, and exponentiation according to established rules. For the given expression \( (x^3)^3 \), you're simplifying by using exponent rules.
This includes organizing and reducing terms, ensuring no further simplifications can be made. Prime factors and like terms are combined through this process. Simplification also assists in more complex operations and helps in solving algebra problems efficiently, providing a clearer understanding of the expressions' behavior.

Powers of a Power

The concept of powers of a power plays a significant role in simplifying expressions with exponents. It occurs when you have an exponent raised to another exponent, as in \((x^3)^3\). Here, the base \(x\) is raised to an exponent, and this result is again subjected to another exponent. The rule \((a^m)^n = a^{m \cdot n}\) simplifies the expression effectively.
This rule simplifies the nested exponents by multiplying them directly. It's a powerful technique for managing complex algebraic expressions, saving time and effort. Understanding powers of a power helps you tackle more sophisticated algebra problems by identifying patterns and processes that underpin algebraic manipulation.

Mathematical Simplification Process

Mathematical simplification is the process of transforming a complicated expression into a simpler and more manageable form. This involves applying rules and techniques, such as the exponentiation rule, in a logical manner. Through simplification, we clearly identify the core elements of an expression and reduce unnecessary complexity. For example, in \( (x^3)^3 \), systematically applying exponent rules leads to the simple form \(x^{27}\).

• Break down the expression
• Identify applicable rules
• Apply rules step-by-step
• Verify simplified results

By following these steps, you ensure a logical flow in your solution. Simplification enhances clarity, making it easier to understand and work with mathematical expressions.