Problem 25 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$2^{-3} \cdot 2^{-4}$$

Step-by-Step Solution

Answer

\(\frac{1}{128}\)

Step 1: Identify the Operation

We need to perform multiplication of like bases, i.e., both terms have the base 2.

Step 2: Apply the Law of Exponents

When multiplying like bases, add the exponents: \(2^{-3} \cdot 2^{-4} = 2^{-3 + (-4)} = 2^{-7}\).

Step 3: Convert to Positive Exponents

To express \(2^{-7}\) with positive exponents, take the reciprocal of the base: \(2^{-7} = \frac{1}{2^7}\).

Step 4: Simplify the Expression

Calculate \(2^7\): \(2^7 = 128\). Therefore, \(\frac{1}{2^7} = \frac{1}{128}\).

Key Concepts

Negative Exponents

Negative exponents are a fundamental aspect of working with exponents in algebra. They indicate the reciprocal of the base raised to the corresponding positive exponent. For example, when you see a term like \(2^{-3}\), it doesn't mean that the number is negative, but instead, it means you should take the reciprocal of \(2^3\).
So, \(2^{-3}\) becomes \(\frac{1}{2^3}\).
Understanding this is crucial because it helps in simplifying expressions and solving equations involving exponents.

• When the exponent is negative, flip the base to its reciprocal form with a positive exponent.
• Use this concept to easily convert negative exponents into positive ones during calculations.

Keeping track of these rules allows accurate and efficient handling of problems involving negative exponents.

Laws of Exponents

The laws of exponents form the backbone of calculations in algebra involving exponents. These laws include rules on how to handle operations like multiplication, division, and raising powers to another power. One of the key laws, especially in the context of the original exercise, is the product of powers property.

This law states: when you multiply two numbers with the same base, you can add their exponents. For instance, \(2^{-3} \cdot 2^{-4}\) uses this law directly: you combine the exponents to get \(2^{-3 + (-4)} = 2^{-7}\).

• **Product of Powers Law:** \(a^m \cdot a^n = a^{m+n}\)
• **Power of a Power Law:** \((a^m)^n = a^{mn}\)
• **Power of a Product Law:** \((ab)^m = a^m \cdot b^m\)

By mastering these laws, you can simplify even complex expressions easily and recognize patterns across many different types of problems.

Multiplication of Exponents

Multiplication of exponents involves processes that may at first seem complicated, but become simple with practice and a good understanding of exponent rules. The process focuses on multiplying terms with like bases by using the laws of exponents, especially focusing on adding exponents when the bases are the same.
Let's see how it works with an example: \(2^{-3} \cdot 2^{-4}\). Both terms have the base 2, so we add their exponents: \(-3\) and \(-4\).
This gives us \(2^{-3 + (-4)} = 2^{-7}\). This simplifies the expression significantly.

• When multiplying terms with the same base, simply add their exponents.
• Be sure to perform any subsequent simplifications to arrive at an expression with positive exponents.

Recognizing that multiplication of exponents is just an application of adding numbers allows you to streamline your approach and solve problems more efficiently.