Problem 71 - Algebra 1

Question

\(2 \cdot 2 \cdot 2 \cdot 2\)

Step-by-Step Solution

Answer

The result of \(2 \cdot 2 \cdot 2 \cdot 2\) is 16

Step 1: Converting Consecutive Multiplications into Exponentiation

Technically, \(2 \cdot 2 \cdot 2 \cdot 2\) means that number 2 is being multiplied by itself 4 times, which is the same as \(2^4\)

Step 2: Solving the Exponentiation

In the exponential form \(2^4\), 2 is the base and 4 is the exponent, means multiply 2 by itself for 4 times. So, \(2 \times 2 \times 2 \times 2 = 16\)

Key Concepts

Multiplication

Multiplication is a fundamental mathematical operation that involves adding a number to itself a certain number of times. For example, if you have the expression 2 multiplied by 4, it means you have two added together four times (2 + 2 + 2 + 2 equals 8). This is much simpler and quicker than adding multiple times.
Here are some key points about multiplication:

• Multiplication is commutative, meaning the order of numbers does not affect the result, i.e., 4 × 2 is the same as 2 × 4.
• Each number in multiplication is called a "factor." If you are multiplying 2 by 3, both 2 and 3 are factors.
• The result of a multiplication operation is called the "product."

Understanding multiplication is crucial as it is the backbone for more advanced operations like exponentiation. When multiplying the same number several times, exponentiation provides a simplified alternative.

Exponents

Exponents offer a way to simplify repeated multiplication of the same number. In mathematics, an exponent denotes how many times a number, known as the base, is multiplied by itself.
For instance, in the expression \(2^4\), 2 is the base, and 4 is the exponent. This indicates that you are multiplying 2 by itself four times. Exponents make it easy to handle large multiplications efficiently.
Some important points about exponents include:

• An exponent expresses repeated multiplication, making it quicker to write and calculate long products.
• Zero as an exponent always results in a base value of 1, such as \(2^0 = 1\).
• Exponents follow their own set of rules, such as \((a^m)^n = a^{mn}\), which means you multiply the exponents.

Exponents are widely used in various fields, such as science and engineering, because they simplify calculations and data management.

Base and Exponent

In the world of exponents, two vital components work together: the base and the exponent. Understanding these components is key to solving and simplifying exponential expressions efficiently.
The **base** is the number that gets multiplied repeatedly. In the expression \(2^4\), the number 2 is the base. It's the number you start with before multiplying.
The **exponent**, on the other hand, indicates the number of times the base is multiplied by itself. In the example \(2^4\), 4 is the exponent which shows how many 2's you multiply.
Remember:

• The base is the primary number of interest, determining the value of the multiplication.
• The exponent provides the repeated count for this base, simplifying long, cumbersome multiplication expressions.
• Both base and exponent are essential to form the complete picture in an exponential expression.

By grasping the concepts of base and exponent, you'll easily manage complex calculations and better understand how exponentiation operates in various mathematical contexts.