Problem 66 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. \(2^{4} \cdot 2^{3}\)

Step-by-Step Solution

Answer

The evaluation of the expression \(2^{4} \cdot 2^{3}\) is 128.

Step 1: Identify the base and the exponents

The expression \(2^{4} \cdot 2^{3}\) has the same base of 2 with two different exponents, 4 and 3 respectively.

Step 2: Apply the product of powers law

As per the product of powers law \(a^{m} \cdot a^{n} = a^{m+n}\), adding the exponents 4 and 3 gives \(2^{4+3}\).

Step 3: Simplify the expression

Evaluating \(2^{7}\) gives the value 128.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a certain power, which is indicated by the exponent. The exponent, placed as a superscript to the right of the base, denotes how many times the base is multiplied by itself. For instance, in the expression \(2^4\), 2 is the base and 4 is the exponent, meaning 2 is multiplied by itself 4 times (2 x 2 x 2 x 2).

The process of exponentiation is a shorthand for repeated multiplication and is used extensively in various fields such as science, engineering, and finance to represent growing patterns or compound phenomena. Simplifying exponential expressions requires understanding the laws of exponents, which are rules that define how to handle exponents in mathematical expressions.

Simplifying Expressions

Simplifying expressions is a process in mathematics where we aim to transform a complex or bulky expression into a simpler form, which is easier to understand or use in calculations. For exponential expressions, this often involves reducing the expression to a single exponent, or to a simpler number known as evaluating the exponent.

This does not mean changing the value of the expression, but rather expressing it in the most straightforward way possible. In our original exercise, simplifying \(2^4 \cdot 2^3\) did not require changing the base or the fundamental operation of exponentiation; it simply involved combining the powers into a single exponent, as dictated by the laws of exponents.

Laws of Exponents

The laws of exponents, also known as the rules of indices, are a set of rules that describe how mathematical expressions with exponents are to be handled. One such law is the product of powers law, which states that when multiplying expressions with the same base, one can add the exponents. Thus, \(a^m \cdot a^n = a^{m+n}\).

Applying this law in the context of the original problem, we combined \(2^4\) and \(2^3\) by adding their exponents, resulting in \(2^{4+3}\). It's this straightforward rule that allows for quick and easy simplification without the need for complex calculations or guesswork. The laws of exponents are foundational in algebra and help in manipulating large numbers, solving equations, and understanding geometric growth, amongst many other applications.