Problem 23 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ (2 a)^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((2a)^3\). Answer: \(8a^3\)

Step 1: Identify the base and exponent

Identify the base and the exponent in the expression \((2a)^3\). Here, the base is \((2a)\) and the exponent is \(3\).

Step 2: Apply the power rule

When the base is a product (i.e. \(2a\)) and is raised to an exponent, the exponent distributes to each factor of the product, according to the power rule: \((ab)^n = a^n b^n\). So, we should multiply each factor of the base \((2a)\) by the exponent \(3\) separately: $$ (2a)^3 = 2^3 \cdot a^3 $$

Step 3: Simplify the expression

Now, simplify the result obtained in Step 2. Calculate \(2^3\) which is \(2 \cdot 2 \cdot 2 = 8\). This gives us the simplified expression: $$ 8a^3 $$ So, the simplified expression for \((2a)^3\) is \(8a^3\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. It is represented as \( a^n \), where \( a \) is the base and \( n \) is the exponent. Exponentiation indicates that the base should be multiplied by itself, as many times as the exponent specifies. For instance, \( 2^3 \) means multiplying 2 by itself 3 times, which equals 8.

This operation is fundamental in various areas, such as algebra and calculus. It simplifies the representation of repeated multiplication, making large calculations more manageable. A key rule to remember with exponents is that they must be non-negative integers in expressions like the one we are addressing here. However, it's essential to understand that exponentiation is still valid for negative or zero exponents under certain rules, although these cases are less common in basic algebra.

Simplifying Expressions

Simplifying expressions with exponents involves using certain rules or properties that make working with these expressions more straightforward. One such rule is the power rule, which states that when multiplying like bases, you add the exponents. For example, \( a^m \cdot a^n = a^{m+n} \). However, when dealing with a power of a product, like in \((ab)^n\), you apply the exponent separately to each factor, resulting in \(a^n \cdot b^n\).

In our given expression \((2a)^3\), simplicity is achieved through the power rule. Here, we break down the product within the parentheses by distributing the exponent 3 to each component of the base, resulting in \(2^3 \cdot a^3\). Simplifying further, we calculate \(2^3\) as 8, giving the simplified form \(8a^3\).

This simplification not only makes the expression easier to handle but also prepares it for further operations, should they be needed. Simplifying expressions is particularly useful for solving algebraic equations and for making calculations easier and less error-prone.

Base and Exponent Identification

To work with expressions involving exponents effectively, it's crucial to correctly identify the base and the exponent. The base is the number or expression that is being multiplied, and the exponent signifies how many times to multiply the base by itself.

In the expression \((2a)^3\), the entire \(2a\) is the base. It's a common mistake to overlook and consider either 2 or \(a\) individually as the base instead of the whole product. The exponent 3 tells us that this base should be multiplied by itself three times: \((2a) \times (2a) \times (2a)\).

Correct identification ensures that applying power rules and other simplification methods go smoothly. Failing to recognize the correct base or exponent can lead to errors in computation, affecting the final results. Thus, practicing clear identification paves the way for accurate and efficient simplification, especially as expressions become more complex with added variables and constants.