The Expression $x^{22}\left( X^7\right)^3$ Is Equivalent To $x^{p}$. What Is The Value Of \$p$[/tex\]?

Introduction

In mathematics, the concept of exponents is crucial in understanding various mathematical operations and expressions. The expression $x^{22}\left( x^7\right)^3$ is a classic example of how exponents can be manipulated to simplify complex expressions. In this article, we will delve into the world of exponents and explore how the given expression is equivalent to $x^{p}$, where $p$ is a value that needs to be determined.

Understanding Exponents

Before we dive into the expression $x^{22}\left( x^7\right)^3$, let's take a moment to understand the concept 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 $x^3$, the exponent $3$ indicates that the base number $x$ should be multiplied by itself three times, resulting in $x \cdot x \cdot x$.

Simplifying the Expression

Now that we have a basic understanding of exponents, let's focus on simplifying the expression $x^{22}\left( x^7\right)^3$. To do this, we need to apply the rule of exponents that states when we multiply two numbers with the same base, we add their exponents. In this case, we have $x^{22}$ and $\left( x^7\right)^3$, both of which have the same base $x$. Therefore, we can add their exponents to get:

$$x^{22}\left( x^7\right)^3 = x^{22+21}$$

Applying the Rule of Exponents

Now that we have simplified the expression to $x^{22+21}$, let's apply the rule of exponents that states when we add two exponents, we multiply the base by itself as many times as the sum of the exponents. In this case, we have $22+21=43$, so we can multiply the base $x$ by itself $43$ times:

$$x^{22+21} = x^{43}$$

Determining the Value of $p$

Now that we have simplified the expression to $x^{43}$, we can see that it is equivalent to $x^{p}$, where $p$ is the value that we need to determine. Since the expression $x^{43}$ is equivalent to $x^{p}$, we can conclude that $p=43$.

Conclusion

In conclusion, the expression $x^{22}\left( x^7\right)^3$ is equivalent to $x^{p}$, where $p$ is the value that we need to determine. By applying the rule of exponents and simplifying the expression, we have shown that $p=43$. This demonstrates the importance of understanding exponents and how they can be manipulated to simplify complex expressions.

Frequently Asked Questions

• What is the value of $p$ in the expression $x^{22}\left( x^7\right)^3$?
• How do we simplify the expression $x^{22}\left( x^7\right)^3$ using the rule of exponents?
• What is the equivalent form of the expression $x^{22}\left( x^7\right)^3$?

Final Answer

The final answer is: $\boxed{43}$

Introduction

In our previous article, we explored the concept of exponents and how the expression $x^{22}\left( x^7\right)^3$ is equivalent to $x^{p}$. We determined that $p=43$ by applying the rule of exponents and simplifying the expression. In this article, we will answer some frequently asked questions related to the expression $x^{22}\left( x^7\right)^3$ and its equivalent form $x^{p}$.

Q&A

Q: What is the value of $p$ in the expression $x^{22}\left( x^7\right)^3$?

A: The value of $p$ in the expression $x^{22}\left( x^7\right)^3$ is $43$. This is because when we simplify the expression using the rule of exponents, we get $x^{22+21}=x^{43}$.

Q: How do we simplify the expression $x^{22}\left( x^7\right)^3$ using the rule of exponents?

A: To simplify the expression $x^{22}\left( x^7\right)^3$ using the rule of exponents, we need to apply the rule that states when we multiply two numbers with the same base, we add their exponents. In this case, we have $x^{22}$ and $\left( x^7\right)^3$, both of which have the same base $x$. Therefore, we can add their exponents to get $x^{22+21}=x^{43}$.

Q: What is the equivalent form of the expression $x^{22}\left( x^7\right)^3$?

A: The equivalent form of the expression $x^{22}\left( x^7\right)^3$ is $x^{p}$, where $p$ is the value that we need to determine. In this case, we have determined that $p=43$, so the equivalent form of the expression is $x^{43}$.

Q: Can we simplify the expression $x^{22}\left( x^7\right)^3$ further?

A: Yes, we can simplify the expression $x^{22}\left( x^7\right)^3$ further by applying the rule of exponents. Since we have already determined that $x^{22}\left( x^7\right)^3=x^{43}$, we can conclude that the expression cannot be simplified further.

Q: What is the importance of understanding exponents in mathematics?

A: Understanding exponents is crucial in mathematics because it allows us to simplify complex expressions and solve problems more efficiently. Exponents are used in various mathematical operations, including multiplication, division, and exponentiation.

Q: How do we apply the rule of exponents in real-world problems?

A: The rule of exponents is applied in real-world problems by simplifying complex expressions and solving problems more efficiently. For example, in finance, exponents are used to calculate compound interest and investment returns. In science, exponents are used to describe the growth and decay of populations and chemical reactions.

Conclusion

In conclusion, the expression $x^{22}\left( x^7\right)^3$ is equivalent to $x^{p}$, where $p$ is the value that we need to determine. By applying the rule of exponents and simplifying the expression, we have shown that $p=43$. We have also answered some frequently asked questions related to the expression $x^{22}\left( x^7\right)^3$ and its equivalent form $x^{p}$.

Frequently Asked Questions

• What is the value of $p$ in the expression $x^{22}\left( x^7\right)^3$?
• How do we simplify the expression $x^{22}\left( x^7\right)^3$ using the rule of exponents?
• What is the equivalent form of the expression $x^{22}\left( x^7\right)^3$?
• Can we simplify the expression $x^{22}\left( x^7\right)^3$ further?
• What is the importance of understanding exponents in mathematics?
• How do we apply the rule of exponents in real-world problems?

Final Answer

The final answer is: $\boxed{43}$