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Chapter 2: Problem 45

Simplify the given expression as completely as possible. $$\left(a^{3}\right)^{2}$$

Short Answer

Expert verified
The simplified expression is \(a^{6}\).

Step by step solution

01

Understand the Problem

The task is to simplify the given expression \(\left(a^{3}\right)^{2}\). This involves using exponent rules to combine the powers.
02

Apply the Power of a Power Rule

According to the power of a power rule, \(\left( x^m \right)^n = x^{m \cdot n}\). Apply this rule to the expression \(\left(a^{3}\right)^{2}\).
03

Multiply the Exponents

Multiply the exponents together: \(a^{3 \cdot 2} = a^{6}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying expressions
Simplifying expressions in algebra involves reducing them to their simplest form. This is done to make the equations easier to work with. When you encounter an expression with exponents, it's important to apply the correct rules to combine and simplify the terms. In the given exercise, we have the expression \(\left(a^{3}\right)^{2}\). To simplify this, we'll need to apply exponent rules correctly.
power of a power rule
The power of a power rule is a fundamental concept in algebra that helps us simplify expressions with exponents. This rule states: \(\left( x^m \right)^n = x^{m \cdot n}\). In other words, when raising an exponentiated term to another power, you multiply the exponents together. In the given exercise \(\left(a^{3}\right)^{2}\), applying this rule involves multiplying the exponents. So, \(3 \cdot 2 = 6\). Therefore, \(\left(a^{3}\right)^{2} = a^{6}\).
multiplying exponents
Multiplying exponents is a key step in simplifying expressions using the power of a power rule. When we apply this rule, we're essentially combining the repeated multiplication of the same base. In the example provided, \(\left(a^{3}\right)^{2}\), we have an exponent of 3 that is raised to the power of 2. By multiplying these exponents (3 and 2), we simplify the expression to \(a^{6}\). This simplification process helps to streamline equations and make them more manageable for further mathematical operations.
Understanding how to multiply exponents correctly is crucial for mastering algebraic expressions.

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