Question

Write the expression in exponential form. $$ 4 x \cdot 4 x \cdot 4 x $$

Short answer

The exponential form of \(4x \cdot 4x \cdot 4x\) is \((4x)^3\).

Step-by-step solution

Identify the Base

First, identify the base in the given multiplication. Here, the base is \(4x\) because every term in the multiplication is the same - \(4x\).

Count the Repeated Terms

Count the number of times the base repeats. The base, \(4x\), repeats three times since we're multiplying \(4x * 4x * 4x\). Hence, the number of times \(4x\) is repeated is the exponent in the exponential form.

Write in Exponential Form

Combine the base and the count to form the exponential expression. The base is \(4x\), and it's repeated 3 times. In exponential notation, repeated multiplication is indicated by a power or exponent, so this could be written as \((4x)^3\).

Key concepts

Base in Exponentiation

Understanding the base in exponentiation is crucial for mastering the concept of powers and indices. The base is the number or expression that is being multiplied by itself a certain number of times. In our exercise, the base is 4x. This base appears in each term of the multiplication, indicating that it is the foundational value in the repeated multiplication process. It's like the building block for constructing a larger expression using exponents.

When writing an expression in exponential form, the base is always written first, followed by an exponent, which indicates how many times the base is multiplied by itself. In our example, the base 4x is used thrice; hence, it becomes the cornerstone of our exponential expression. Recognizing the base is the first step in converting a repeated multiplication into its compact exponential form.

Exponents

Exponents, also known as powers, are a shorthand notation to represent repeated multiplication of the same number or expression. When you see \[\begin{equation}(4x)^3,\end{equation}\] the '3' here is called the exponent. The exponent indicates that the base, 4x, is used as a factor three times. Instead of writing out the long form, 4x • 4x • 4x, we use the exponent to simplify and convey this information neatly and concisely.

The power of exponents lies in their ability to express very large numbers or equations in a much more manageable way. For example, 10 raised to the power of 6, written as 10^6, is a lot easier to read and write than 1,000,000 (one million), which illustrates the same idea through repeated multiplication.

Repeated Multiplication

Repeated multiplication is at the heart of understanding exponentiation. When we repeat a multiplication process, such as 4x • 4x • 4x, it becomes cumbersome to write as the number of repetitions increases. Imagine multiplying 4x by itself 10 times; it would be quite a long equation! That's where exponents come into play, offering a more efficient way to represent such operations.

Repeated multiplication is always assumed to be of the same base, which builds a potent computational rule: instead of elongating the expression, we can consolidate it using an exponent to indicate how many times the base is used. So, when simplifying the expression 4x • 4x • 4x, we've already identified the base and counted that it's used three times, writing it as (4x)^3 is a perfect illustration of using exponents to represent repeated multiplication in a compact and understandable form.