Problem 32 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ 4^{-1} \cdot 4^{-1} $$

Step-by-Step Solution

Answer

The evaluation of the expression \(4^{-1} \cdot 4^{-1}\) without using a calculator is 1/16.

Step 1: Identify Base and Exponent

The base in this expression is 4 and the exponent in both cases is -1. Exponentiation is a mathematical operation, denoted as \(b^n\), involving two numbers, the base \(b\) and the exponent or power \(n\). When the base is negative, the effect is of taking the reciprocal of the base when it is to the positive power.

Step 2: Apply the Exponent Rule

When multiplying expressions with the same base, the exponents should be added. Therefore, \(4^{-1} \cdot 4^{-1} = 4^{-1 + -1} = 4^{-2}\).

Step 3: Interpret Negative Exponents

A negative exponent means that we take the reciprocal of the base to the power of the absolute value of the exponent. This means \(4^{-2} = 1/4^{2} = 1/16\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves two key elements: a base and an exponent. You might see it written as \( b^n \), where \( b \) is the base and \( n \) is the exponent.
This operation tells us how many times to multiply the base by itself. For instance, \( 4^2 \) means multiplying 4 by itself, which is \( 4 \times 4 = 16 \).
However, when dealing with negative exponents, such as in \( 4^{-1} \), the rule changes a bit. Rather than multiplying, you take the reciprocal of the base raised to the positive of that exponent. So, \( 4^{-1} \) becomes \( \frac{1}{4^1} \), which equals \( \frac{1}{4} \).
Exponentiation is crucial in mathematics as it simplifies repeated multiplication, especially with larger numbers.

Base and Exponent

Understanding the components of exponentiation—base and exponent—is essential to solving expressions like \( 4^{-1} \).
In the expression \( b^n \):

• The **base** (\( b \)) is the number you are multiplying. In our example, it's 4.
• The **exponent** (\( n \)) tells us how many times to multiply the base. If the exponent is negative, it suggests taking the reciprocal.

Let's look closer at the expression \( 4^{-1} \). Here, 4 is the base, and -1 is the exponent. The negative exponent indicates that we need the reciprocal, turning the expression into \( \frac{1}{4} \).
This understanding helps in working through calculations systematically and is foundational for tackling more complex math problems.

Multiplying Exponents

When multiplying exponents with the same base, you apply a straightforward rule: **add the exponents**.
This principle helps simplify expressions. Say you have \( 4^{-1} \cdot 4^{-1} \). Here, the bases are the same (both 4), so you add the exponents:
\[ 4^{-1} \times 4^{-1} = 4^{-1 + (-1)} = 4^{-2} \]
Now, interpreting this newly formed exponent \(-2\): it suggests finding the reciprocal and squaring it. Therefore, \( 4^{-2} \) becomes \( \frac{1}{4^2} = \frac{1}{16} \).
By using this rule, multiplying similar bases with exponents becomes manageable, allowing you to simplify expressions quickly. This technique is especially handy across various branches of math and science.