2. Which Number Can Be Placed In The Box To Make The Statement True?$\[ \frac{p^{-14}}{p^{-2}} \cdot P^? = P^{21} \\]A. 3 B. 14 C. 33 D. 37

Introduction

In this article, we will delve into the world of exponential equations and explore a specific problem that requires us to find the correct number to place in a box to make a statement true. The statement involves a complex exponential expression, and our goal is to simplify it and identify the missing value. We will break down the problem step by step, using the properties of exponents to arrive at the solution.

Understanding Exponents

Before we dive into the problem, let's quickly review the basics of exponents. An exponent is a small number that is placed above and to the right of a base number, indicating how many times the base number should be multiplied by itself. For example, in the expression $a^b$, the base is $a$ and the exponent is $b$. When we multiply two numbers with the same base, we add their exponents. For instance, $a^m \cdot a^n = a^{m+n}$.

The Problem

The problem we are given is:

$\frac{p^{-14}}{p^{-2}} \cdot p^? = p^{21}$

Our task is to find the missing value in the box, denoted by $p^?$. To do this, we need to simplify the expression on the left-hand side and equate it to the expression on the right-hand side.

Simplifying the Expression

Let's start by simplifying the fraction on the left-hand side:

$\frac{p^{-14}}{p^{-2}} = p^{-14} \cdot p^2 = p^{-14+2} = p^{-12}$

Now, we can rewrite the original expression as:

$p^{-12} \cdot p^? = p^{21}$

Using Exponent Properties

We know that when we multiply two numbers with the same base, we add their exponents. Therefore, we can rewrite the expression as:

$p^{-12+?} = p^{21}$

This implies that the exponents on both sides of the equation must be equal:

$-12+? = 21$

Solving for the Missing Value

Now, we can solve for the missing value by isolating the variable $?$. We can do this by adding $12$ to both sides of the equation:

$? = 21 + 12$

$? = 33$

Therefore, the missing value in the box is $33$.

Conclusion

In this article, we have solved a complex exponential equation by simplifying the expression and using the properties of exponents. We have shown that the missing value in the box is $33$, which makes the statement true. This problem requires a deep understanding of exponents and their properties, as well as the ability to simplify complex expressions. By following the steps outlined in this article, you should be able to solve similar problems and become more confident in your ability to work with exponential equations.

Answer

The correct answer is:

• C. 33

Discussion

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, $2^3$ is an exponential expression, where $2$ is the base and $3$ is the exponent.

Q: How do I simplify an exponential expression?

A: To simplify an exponential expression, you can use the properties of exponents. For example, if you have $a^m \cdot a^n$, you can add the exponents to get $a^{m+n}$. You can also use the rule $a^{-n} = \frac{1}{a^n}$ to simplify negative exponents.

Q: What is the rule for multiplying exponential expressions with the same base?

A: When you multiply two exponential expressions with the same base, you add the exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable. You can do this by using the properties of exponents to simplify the equation and then solving for the variable.

Q: What is the difference between an exponential equation and a linear equation?

A: An exponential equation involves an exponential expression, while a linear equation involves a linear expression. For example, $2^3$ is an exponential expression, while $2x + 3$ is a linear expression.

Q: Can you give an example of an exponential equation?

A: Yes, here is an example of an exponential equation:

$\frac{p^{-14}}{p^{-2}} \cdot p^? = p^{21}$

This equation involves an exponential expression and requires the use of the properties of exponents to simplify and solve.

Q: How do I know which base to use in an exponential equation?

A: The base of an exponential equation is usually a variable or a constant. In the example above, the base is $p$. You can choose any base that makes the equation easier to solve.

Q: Can you explain the concept of negative exponents?

A: Yes, a negative exponent is a way of expressing a fraction. For example, $a^{-n} = \frac{1}{a^n}$. This means that $a^{-n}$ is equal to the reciprocal of $a^n$.

Q: How do I handle negative exponents in an exponential equation?

A: To handle negative exponents in an exponential equation, you can use the rule $a^{-n} = \frac{1}{a^n}$. This allows you to simplify the equation and solve for the variable.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is:

• C. 33

We hope this Q&A article has helped you understand exponential equations and how to solve them. If you have any further questions, please don't hesitate to ask.