Problem 15 - Algebra 1: Concepts and Skills

Question

Write the expression in exponential form. nine to the fifth power

Step-by-Step Solution

Answer

\(9^5\)

Step 1: Identify the base

In this expression, the base number is 'nine'. This is the number that's being raised to a power.

Step 2: Identify the power

The power or exponent is 'fifth'. This tells us how many times the base is to be multiplied by itself.

Step 3: Write in exponential form

The base number comes first, followed by the caret symbol (^), then the exponent. Therefore, 'nine to the fifth power' is written in exponential form as \(9^5\).

Key Concepts

Base Number in Exponentiation

The base number in exponentiation is fundamental to understanding expressions in exponential form. It is the number that will be multiplied by itself a specified number of times as indicated by the exponent. In this case, our base number is "nine." It’s the starting point or foundational element in exponential calculations.

It acts like the root of a plant, where the number itself branches out as it multiplies according to the exponent number.

• In our scenario, since the base is nine, all calculations will revolve around this number.
• Being aware of the base number allows you to set up the problem correctly when multiplying.
• The base number is always noted first when moving to exponential or scientific notation.

Understanding the role of the base number helps simplify and accurately perform tasks involving exponential equations.

Exponent or Power

The exponent, often called the power, is equally critical in understanding exponential notation. It tells you how many times the base should be multiplied by itself.

In our example, the expression is "nine to the fifth power," where five is the exponent. To visualize, think of the exponent as a signal for repeated actions of multiplication:

• The number five indicates multiplying nine by itself a total of five times.
• It determines the "depth" or the "intensity" of multiplication for the base number.
• An exponent of '1' means the base remains as it is, while '0' would mean the base equals one.

This exponent's job is to quickly communicate the number of repetitions needed, bypassing lengthy multiplication steps.

Writing Expressions in Exponential Form

Writing expressions in exponential form simplifies complex multiplication tasks. It provides an efficient shorthand way to represent repeated multiplication.

In our problem, the expression "nine to the fifth power" is written as \(9^5\). This method uses a base number with an attached exponent, indicating the multiplication extent.

• The base, nine, begins the expression.
• The caret symbol (^) is used to separate the base from the exponent number, which follows.
• This computed shorthand allows easier adjustments in computations, especially for high-power exponents.

Transitioning to exponential form is beneficial, not just for ease, but for greater clarity and consistency in advanced mathematical operations.