Problem 15 - College Algebra

Question

Evaluate each exponential expression in Exercises 1–22. $$\left(2^{2}\right)^{3}$$

Step-by-Step Solution

Answer

The result of the expression \( \left(2^{2}\right)^{3} \) is 64.

Step 1: Begin with innermost layer paretheses first

First, evaluate the expression inside the parentheses. \(2^{2}\) means multiplying 2 by itself, which results in 4.

Step 2: Apply next exponent

Then using the result from Step 1, calculate \(4^{3}\). Here, we multiply 4 by itself two more times, which gives 64.

Key Concepts

College Algebra

In college algebra, students delve into more advanced techniques in algebra that are foundational for higher math and real-world problem solving. Understanding the properties of exponents is a significant component of college algebra.

In the context of the given problem, evaluating exponential expressions requires a solid grasp of the rules for manipulating these expressions. College algebra not only helps students to handle numbers and simple algebraic structures but also sets the stage for more complex mathematical concepts encountered in calculus, linear algebra, and beyond.

The exercise provided, evaluating the expression \( (2^{2})^{3} \), demands an understanding of not just arithmetic but how to apply specific algebraic rules related to powers and exponentiation—a skill highly relevant in both theoretical and practical applications.

Exponent Rules

Exponent rules or laws of exponents are critical tools in simplifying and evaluating expressions involving powers. Here are some fundamental rules that students must master:

• The 'Product Rule' for handling expressions like \( x^m \cdot x^n \) where the bases are the same. You add the exponents, resulting in \( x^{m+n} \).
• The 'Power to a Power Rule' is illustrated by \( (x^m)^n \), where you multiply the exponents, resulting in \( x^{m\cdot n} \).
• The 'Power of a Product Rule' tells us that \( (xy)^n = x^n \cdot y^n \).
• The 'Quotient Rule' for expressions like \( \frac{x^m}{x^n} \) involves subtracting the exponents, giving \( x^{m-n} \).
• The 'Zero Exponent Rule' states that any nonzero base raised to the power of zero equals one, \( x^0 = 1 \).

Understanding and applying these exponent rules is essential for mathematical computations involving powers. The provided exercise uses the 'Power to a Power Rule' to simplify \( (2^{2})^3 \), simplifying to \( 2^{2\cdot3} \) or \( 2^6 \) before calculating the final value.

Mathematical Computations

Mathematical computations lie at the heart of solving algebraic problems. They are the step-by-step procedures that allow one to derive the solution to a numerical or algebraic expression.

In our case, evaluating exponential expressions requires computation skills that include not simply multiplying numbers but understanding the iterative process underpinning exponentiation. After applying the relevant exponent rules, the final computational step is to calculate the end value.

When students approach a problem, they typically break it down into understandable parts, compute each part as needed, and then synthesize these parts to find the solution. In the example exercise, computation skills are applied by first calculating \(2^2 = 4\), then raising this result to a power of three to get the final answer, \(4^3 = 64\).