Write The Following Exponential Expression In Expanded Form: $2^{15}$.Enter Your Answer In The Following Format: A \times A \times A, Using The Multiplication Symbol On The Math Keypad.Provide Your Answer Below:

Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to expand them is crucial for solving various mathematical problems. In this article, we will focus on writing the exponential expression $2^{15}$ in expanded form. We will use the multiplication symbol on the math keypad to represent the repeated multiplication of the base number.

What is an Exponential Expression?

An exponential expression is a mathematical expression that represents a quantity that is raised to a power. It is written in the form $a^{n}$, where $a$ is the base number and $n$ is the exponent. The exponent indicates how many times the base number is multiplied by itself.

Expanded Form of Exponential Expressions

To write an exponential expression in expanded form, we need to multiply the base number by itself as many times as indicated by the exponent. In the case of $2^{15}$, we need to multiply 2 by itself 15 times.

Step-by-Step Solution

To expand the exponential expression $2^{15}$, we will follow these steps:

1. Understand the Exponent: The exponent 15 indicates that we need to multiply 2 by itself 15 times.
2. Write the Base Number: The base number is 2.
3. Multiply the Base Number by Itself: We will multiply 2 by itself 15 times, using the multiplication symbol on the math keypad.

Expanded Form of $2^{15}$

Using the multiplication symbol on the math keypad, we can write the expanded form of $2^{15}$ as follows:

2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Simplifying the Expanded Form

To simplify the expanded form, we can group the repeated multiplication of the base number. We can group 2 × 2 as 4, 4 × 4 as 16, and so on.

2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) = 2 × 4 × 16 × 256 × 65536

Final Answer

The expanded form of $2^{15}$ is:

2 × 4 × 16 × 256 × 65536

Conclusion

In this article, we have learned how to write the exponential expression $2^{15}$ in expanded form. We have used the multiplication symbol on the math keypad to represent the repeated multiplication of the base number. We have also simplified the expanded form by grouping the repeated multiplication of the base number. This article provides a comprehensive guide to understanding and expanding exponential expressions.

Frequently Asked Questions

Q: What is an exponential expression?

A: An exponential expression is a mathematical expression that represents a quantity that is raised to a power. It is written in the form $a^{n}$, where $a$ is the base number and $n$ is the exponent.

Q: How do I write an exponential expression in expanded form?

A: To write an exponential expression in expanded form, you need to multiply the base number by itself as many times as indicated by the exponent.

Q: What is the expanded form of $2^{15}$?

A: The expanded form of $2^{15}$ is:

2 × 4 × 16 × 256 × 65536

Q: How do I simplify the expanded form of an exponential expression?

Q&A: Expanded Form of Exponential Expressions

Q: What is an exponential expression?

A: An exponential expression is a mathematical expression that represents a quantity that is raised to a power. It is written in the form $a^{n}$, where $a$ is the base number and $n$ is the exponent.

Q: How do I write an exponential expression in expanded form?

A: To write an exponential expression in expanded form, you need to multiply the base number by itself as many times as indicated by the exponent.

Q: What is the expanded form of $2^{15}$?

A: The expanded form of $2^{15}$ is:

2 × 4 × 16 × 256 × 65536

Q: How do I simplify the expanded form of an exponential expression?

A: You can simplify the expanded form by grouping the repeated multiplication of the base number.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is a mathematical expression that represents a quantity that is raised to a power, while a polynomial expression is a mathematical expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication.

Q: Can I use a calculator to find the expanded form of an exponential expression?

A: Yes, you can use a calculator to find the expanded form of an exponential expression. However, it is also important to understand the concept of exponential expressions and how to write them in expanded form.

Q: How do I determine the exponent of an exponential expression?

A: The exponent of an exponential expression is the number that is raised to the power of the base number. For example, in the expression $2^{15}$, the exponent is 15.

Q: Can I have a negative exponent in an exponential expression?

A: Yes, you can have a negative exponent in an exponential expression. A negative exponent indicates that the base number is raised to a power that is the reciprocal of the exponent.

Q: How do I handle negative exponents in exponential expressions?

A: To handle negative exponents, you can rewrite the expression with a positive exponent by taking the reciprocal of the base number.

Q: Can I have a fractional exponent in an exponential expression?

A: Yes, you can have a fractional exponent in an exponential expression. A fractional exponent indicates that the base number is raised to a power that is a fraction of the exponent.

Q: How do I handle fractional exponents in exponential expressions?

A: To handle fractional exponents, you can rewrite the expression with a positive exponent by taking the square root of the base number.

Q: What is the importance of understanding exponential expressions?

A: Understanding exponential expressions is important because it helps you to solve a wide range of mathematical problems, from simple arithmetic to complex calculus.

Q: Can I use exponential expressions in real-world applications?

A: Yes, you can use exponential expressions in real-world applications, such as finance, science, and engineering.

Q: How do I apply exponential expressions in real-world applications?

A: To apply exponential expressions in real-world applications, you need to understand the concept of exponential expressions and how to use them to model real-world phenomena.

Conclusion

In this article, we have provided a comprehensive guide to understanding and expanding exponential expressions. We have also answered frequently asked questions about exponential expressions and provided examples of how to apply them in real-world applications.