Problem 45 - Algebra 1

Question

Write the expression in exponential form. \(x\) raised to the sixth power

Step-by-Step Solution

Answer

\(x^6\)

Step 1: Understanding Exponential Form

Exponential form combines three components: base, exponent and equal value. The base is the number that is being multiplied. The exponent, located to the upper right of the base, tells how many times the base is multiplied by itself. The equal value is the result of that multiplication.

Step 2: Apply the understanding to the given problem

In this problem, \(x\) is the base and the sixth power represents the exponent. So, putting it together we get \(x^6\)

Key Concepts

Base and Exponent

In mathematics, when we talk about exponential form, we are referring to a way of expressing repeated multiplication of a number by itself. An important part of this expression is understanding two key components: the base and the exponent. The base is the number that is being multiplied. For example, if we have a number like 2 in an expression, and it's going to be multiplied multiple times, this number 2 is our base.

The exponent is a small number written to the upper right of the base. This small number indicates how many times the base is used as a factor in the multiplication. Think of it as a kind of shorthand. Instead of writing out 2 multiplied by itself, say, six times (2 × 2 × 2 × 2 × 2 × 2), we can write it in exponential form as \(2^6\).

In this form, 2 is the base and 6 is the exponent, demonstrating that 2 is multiplied by itself six times. This makes our mathematical expressions cleaner and easier to understand, especially when dealing with large numbers."

Power of a Number

The power of a number essentially tells us how many times the base is used in the multiplication. It is the result of the base being raised to an exponent. In our previous example of \(2^6\), we say that we are raising 2 to the sixth power. This simply means that 2 is used as a factor six times. When a number is raised to a power, the outcome is called a power of that number. The term sometimes causes confusion, but just think of it as an expression of repeated multiplication. It’s like saying a number multiplied through its own power.

• If a number is raised to the power of zero (e.g., \(x^0\)), the result is 1, for any base \(x\), except when the base is zero.
• If a number is raised to the power of one (e.g., \(x^1\)), the result is just the base itself.

Power of a number is a powerful (pun intended) tool in mathematics, simplifying expressions and calculations significantly.

Mathematical Notation

Mathematical notation is a universal language used to represent numbers, formulas, and expressions in a clear and concise manner. It's a system that mathematicians around the world use to communicate complex ideas efficiently. Exponential notation, like \(x^n\), is one part of this language.

When expressing numbers or calculations in exponential form, we can convey a lot of information with only a few symbols. This notation saves time and space when writing and helps avoid errors in complex computations. For example, instead of the long multiplication \(x \times x \times x \times x \times x \times x\), we can simply use \(x^6\).

• The base \(x\) signifies the number being multiplied.
• The exponent 6 indicates how many times the base is used as a factor.

In mathematical notation, clarity and efficiency are key. It allows us to work with numbers effectively, ensuring that our mathematical expressions are both elegant and error-free.