Problem 83 - Prealgebra

Question

Perform the indicated operations. $$4^{3}$$

Step-by-Step Solution

Answer

The result is 64.

Step 1: Understand the Exponential Notation

In the expression \(4^3\), the number 4 is the base, and 3 is the exponent. This means that 4 is multiplied by itself a total of 3 times.

Step 2: Multiply the Base

Calculate \(4 imes 4\), which equals 16. This represents multiplying the base, 4, by itself once.

Step 3: Multiply Again by the Base

Now multiply the result from Step 2 by the base again: \(16 imes 4 = 64\). This step completes the operation of raising 4 to the power of 3.

Key Concepts

Exponential Notation

Exponential notation is a mathematical method of representing numbers using a base and an exponent. It offers a compact way to express large computations and repeated multiplications. The notation involves writing a number as a power. For instance, in the expression \(4^3\), this signifies that the number 4 is to be used as a factor in multiplication three times.
Using exponential notation simplifies expressions and calculations. Instead of writing long multiplication sequences, you consolidate it into a neat format that highlights the base number and how many times it is used as a factor. This method of writing is fundamental in mathematics, making calculations involving powers more intuitive and organized.

Base and Exponent

In exponential notation, understanding the roles of the base and the exponent is crucial. The base is the number that is being multiplied, while the exponent indicates how many times the base is multiplied by itself.
In \(4^3\):

• The number 4 is called the base. It is the number to be multiplied.
• The number 3 is the exponent. It specifies that the base 4 is to be multiplied by itself 3 times.

This communication between the base and exponent allows us to perform complex calculations efficiently. A change in either number significantly alters the outcome. For example, \(4^3\) is different from \(3^4\), because even though the numbers are the same, their position in the notation changes the calculation and the resulting product.

Power of a Number

The power of a number essentially answers the question: how many times is the base number multiplied by itself? When raised to a power, the base undergoes a consistent multiplication process.
For example, \(4^3\), also read as "four to the third power," means:

• Start with the base, 4.
• Multiply it by itself: \(4 \times 4 = 16\).
• Then, take the result and multiply by the base again: \(16 \times 4 = 64\).

When the power is a positive integer, as in most cases encountered in basic mathematics, this process of repeated multiplication yields the power of the number. Understanding "power" helps in comprehending higher-order applications in algebra, calculus, and beyond.