Problem 19 - Algebra 1

Question

Write the expression in exponential form. nine to the \(y\) th power

Step-by-Step Solution

Answer

The exponential form of 'nine to the \( y \) th power' is \( 9^y \).

Step 1: Identify the base and exponent

In our problem, 'nine' is the base and \( y \) is the exponent. By convention, 'to the power of' or 'raised to' means that the number before it is the base and the number after it is the exponent.

Step 2: Write in exponential form

Exponential form is usually written as base^exponent. So 'nine to the \( y \) th power' should be written as 9^ \( y \).

Key Concepts

Base and Exponent

Understanding the concept of base and exponent is essential in mastering exponential expressions. The base is the number that is being multiplied by itself, and the exponent denotes how many times the base is used as a factor.

For example, when we say 'nine to the yth power' in our exercise, the number nine is the base. This means nine is the value that will be multiplied by itself. The letter y represents the exponent, indicating the number of times nine is to be used in the multiplication.

It's important to note that exponents are not limited to positive integers. They can be negative, which would denote repeated division instead of multiplication, or even fractional, indicating roots. However, in our exercise, we focus on y as a positive integer exponent for simplicity.

Writing Exponents

Writing numbers in exponential form is a way to simplify how we visualize and calculate large figures or repeated multiplications. This form is especially useful in higher mathematics where expressions can become quite complex.

To convert a verbal statement like 'nine to the yth power' into mathematical notation, we use the exponentiation operator '^'. This caret symbol indicates the power to which the base is raised. As per the step-by-step solution, it is written as 9^y. Here, 9 is the base and y is the exponent.

It's important for learners to remember to write the base as a numeral and the exponent as a superscript immediately after the base. This concise notation makes it clearer and easier to understand than writing out the long form of repeated multiplication.

Exponential Expressions

Exponential expressions represent repeated multiplication of a base number. They form the core of exponential notation and have various applications across different fields such as science, finance, and technology.

In our textbook exercise, the exponential expression would be 9^y, clearly indicating that nine (the base) is to be multiplied by itself y times (the exponent). However, an understanding of exponential expressions doesn't end here. Students should grasp that these expressions can be manipulated according to the laws of exponents, such as the product of powers rule, the power of a power rule, and the power of a product rule.

As students progress, they'll encounter more complex expressions with exponents, including those with variables and those involving exponential growth and decay. It's critical to have a solid understanding of the basic concepts to tackle these more advanced topics.