Problem 39 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ \left(8 x^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression \((8x^3)^2\) is \(64x^6\).

Step 1: Exponent Rule Application

Use the exponent rule \((a^n)^m=a^{n*m}\) where 'a' is the base, 'n' is the first exponent and 'm' the second exponent. In the given, \(a=8x^3\) and \(m=2\) . So the expression becomes \(8^2*(x^3)^2\).

Step 2: Evaluation of Exponents

This evaluation step involves two parts. First, evaluate \(8^2\) which results in 64. Second, apply the exponent rule again on \((x^3)^2\) which results in \(x^6\).

Step 3: Final Evaluation

Multiply the results together to get the final simplified expression which is \(64*x^6\).

Key Concepts

Exponent Rules

Working with exponential expressions requires an understanding of the rules that govern how to manipulate exponents. These exponent rules help in simplifying complex expressions and solving equations more efficiently. One basic rule is the power of a power rule, which states that when you have an exponent raised to another exponent, like \( (a^n)^m \), you can simplify it by multiplying the exponents, resulting in \( a^{n*m} \). This is exactly what is applied in Step 1, where \( 8x^3 \) is our base \( a \), raised to the power of 2. By multiplying the exponents, we get \( 8^2 \) and \( (x^3)^2 \) separately. Understanding such rules is key to simplifying exponential expressions effortlessly.

Evaluation of Exponents

Once the exponent rules are applied, the next step involves evaluating the expression with the newly formed exponents. To evaluate an expression like \( 8^2 \) or \( (x^3)^2 \) means to calculate its value. Calculating \( 8^2 \) is straightforward and results in 64 because 8 multiplied by itself is 64. However, evaluating \( (x^3)^2 \) requires a bit more understanding of exponents. Here, the exponent of 2 acts on the entire \( x^3 \) term, and according to our exponent rules, this gives us \( x^{3*2} \) or \( x^6 \)—meaning that \( x \) is multiplied by itself a total of six times. This step is fundamental for turning complex expressions into their simplest form.

Algebraic Simplification

The final step in the process of simplifying exponential expressions is algebraic simplification. This brings together the results of evaluating the base and the exponent. In our example, this meant taking the evaluated base of 64 and the evaluated exponent result of \( x^6 \) and combining them to produce the final simplified form of \( 64*x^6 \). The goal of algebraic simplification is clarity and conciseness, aiming to express the solution in the simplest terms possible for easy understanding and application in further mathematical operations or applications. Mastery in algebraic simplification not only improves problem-solving skills but also builds a strong foundation for advanced algebraic concepts.