Question

Write the expression in exponential form. $$6 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x$$

Short answer

The exponential form of \(6 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x\) is \(6x^6\).

Step-by-step solution

Identify the Base

Observe what number or symbol is being multiplied by itself repeatedly. Here, the base appears to be \(x\).

Count the Number of Times the Base is Repeated

Count how many times \(x\) is being multiplied. Here, \(x\) appears 6 times.

Write in Exponential Form

Write the base once followed by an exponent that equals the number of times the base is repeated in the multiplication. That leads us to the expression \(x^6\).

Key concepts

Understanding the Base in Exponential Expressions

When we talk about the base in exponential expressions, we're referring to the number or variable that is being multiplied by itself. In the context of the exercise, the base is the letter 'x.' The idea of a base helps us simplify repeated multiplication; instead of writing 'x' multiple times, we identify 'x' as the base and use exponent notation to convey the same information more concisely.

For instance, when we encounter an expression like
\[\[\begin{align*}6\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\end{align*}\]\]
we can clearly see that 'x' is the common factor being multiplied together repeatedly. The presence of a base is a key element in exponential notation because it anchors the expression and indicates what's being raised to a power. Understanding the base is the first step in translating a repeated multiplication into an exponential form.

Writing Expressions with Exponents

Expressions with exponents are a shorthand way to represent repeated multiplication of the same factor. Writing an expression with exponents not only simplifies notation but also facilitates easier manipulation for algebraic operations. In our example,
\[\[\begin{align*}6\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\end{align*}\]\]
the factor 'x' appears multiple times, and we can express this more succinctly using an exponent.

To convert a multiplication expression like this into one with exponents, we count how many times the base (in this case, 'x') appears. That count becomes our exponent, a small number written just above and to the right of the base. In this exercise, 'x' is repeated 6 times, so the expression in exponential form is written as \[\[\begin{align*}x^6\end{align*}\]\]
This is far easier to write and read, especially with larger numbers.

Grasping Exponent Notation

Exponent notation is a mathematical shortcut that helps us represent large numbers or expressions in a compact form. It consists of two parts: the base and the exponent. The base is the number or variable that is being multiplied by itself, and the exponent tells us how many times the base is to be used in the multiplication.

So, when we write \[\[\begin{align*}x^6\end{align*}\]\]
the 'x' is the base and '6' is the exponent. This tells us that 'x' should be multiplied by itself a total of six times. Exponent notation is not only limited to algebra; it's also used in various scientific fields to represent very large or very small numbers, like in scientific notation.

It's essential to remember that the base can be any number or expression, and the exponent is always a number that indicates the count of the bases. In some cases, an exponent can also be a variable, leading to more advanced algebraic concepts. Mastering exponent notation is fundamental to succeeding in math as it's the groundwork for higher-level operations involving powers, roots, and logarithms.