Which Expressions Can Be Simplified As $\frac{1}{p^{18}}$? Check All That Apply.A. $\left(p^{-4}\right)^4$B. $\left(p^8\right)^{-2}$C. $\left(p^0\right)^{-16}$D. $\left(p^{16}\right)^{-1}$E.

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Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the expressions that can be simplified as $\frac{1}{p^{18}}$. We will examine each option carefully and provide step-by-step solutions to determine which expressions meet the criteria.

Understanding Exponential Properties

Before we dive into the solutions, it's essential to understand the properties of exponents. The power of a power property states that $(a^m)^n = a^{mn}$. This property will be instrumental in simplifying the given expressions.

Option A: $\left(p^{-4}\right)^4$

Let's start by simplifying option A using the power of a power property:

$$\left(p^{-4}\right)^4 = p^{-4 \cdot 4} = p^{-16}$$

To express this as $\frac{1}{p^{18}}$, we need to rewrite it as a fraction with a negative exponent:

$$p^{-16} = \frac{1}{p^{16}}$$

This is not equal to $\frac{1}{p^{18}}$, so option A is not a correct solution.

Option B: $\left(p^8\right)^{-2}$

Next, let's simplify option B using the power of a power property:

$$\left(p^8\right)^{-2} = p^{8 \cdot -2} = p^{-16}$$

To express this as $\frac{1}{p^{18}}$, we need to rewrite it as a fraction with a negative exponent:

$$p^{-16} = \frac{1}{p^{16}}$$

This is not equal to $\frac{1}{p^{18}}$, so option B is not a correct solution.

Option C: $\left(p^0\right)^{-16}$

Now, let's simplify option C using the power of a power property:

$$\left(p^0\right)^{-16} = p^{0 \cdot -16} = p^0$$

Since any non-zero number raised to the power of 0 is equal to 1, we have:

$$p^0 = 1$$

This is not equal to $\frac{1}{p^{18}}$, so option C is not a correct solution.

Option D: $\left(p^{16}\right)^{-1}$

Next, let's simplify option D using the power of a power property:

$$\left(p^{16}\right)^{-1} = p^{16 \cdot -1} = p^{-16}$$

To express this as $\frac{1}{p^{18}}$, we need to rewrite it as a fraction with a negative exponent:

$$p^{-16} = \frac{1}{p^{16}}$$

This is not equal to $\frac{1}{p^{18}}$, so option D is not a correct solution.

Option E: $\left(p^{18}\right)^{-1}$

Finally, let's simplify option E using the power of a power property:

$$\left(p^{18}\right)^{-1} = p^{18 \cdot -1} = p^{-18}$$

To express this as $\frac{1}{p^{18}}$, we need to rewrite it as a fraction with a negative exponent:

$$p^{-18} = \frac{1}{p^{18}}$$

This is equal to $\frac{1}{p^{18}}$, so option E is a correct solution.

Conclusion

In conclusion, only option E, $\left(p^{18}\right)^{-1}$, can be simplified as $\frac{1}{p^{18}}$. The other options do not meet the criteria, and their simplifications are not equal to $\frac{1}{p^{18}}$. By understanding the properties of exponents and applying them to the given expressions, we can determine which expressions can be simplified as $\frac{1}{p^{18}}$.

Additional Tips and Tricks

• When simplifying exponential expressions, always start by applying the power of a power property.
• Be careful when rewriting expressions with negative exponents as fractions.
• Make sure to check your work by plugging in values or using a calculator to verify your solutions.

Q: What is the power of a power property?

A: The power of a power property states that $(a^m)^n = a^{mn}$. This means that when you raise a power to another power, you multiply the exponents.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can rewrite it as a fraction with a positive exponent. For example, $a^{-m} = \frac{1}{a^m}$.

Q: What is the difference between $p^{-16}$ and $\frac{1}{p^{16}}$?

A: $p^{-16}$ is equal to $\frac{1}{p^{16}}$, but they are not the same thing. $p^{-16}$ is an expression with a negative exponent, while $\frac{1}{p^{16}}$ is a fraction with a positive exponent.

Q: Can I simplify an expression with a zero exponent?

A: Yes, any non-zero number raised to the power of 0 is equal to 1. For example, $p^0 = 1$.

Q: How do I know which option is correct?

A: To determine which option is correct, you need to simplify each expression using the power of a power property and check if it equals $\frac{1}{p^{18}}$.

Q: What if I get a different answer than the one in the solution?

A: If you get a different answer than the one in the solution, it's possible that you made a mistake in your calculations. Double-check your work and make sure you applied the power of a power property correctly.

Q: Can I use a calculator to check my work?

A: Yes, you can use a calculator to check your work. However, make sure to enter the expression correctly and use the correct order of operations.

Q: What if I'm still having trouble simplifying an expression?

A: If you're still having trouble simplifying an expression, try breaking it down into smaller parts and simplifying each part separately. You can also ask a teacher or tutor for help.

Q: Are there any other properties of exponents that I should know about?

A: Yes, there are several other properties of exponents that you should know about, including:

• The product of powers property: $a^m \cdot a^n = a^{m+n}$
• The quotient of powers property: $\frac{a^m}{a^n} = a^{m-n}$
• The power of a product property: $(ab)^m = a^m \cdot b^m$

These properties can help you simplify more complex expressions and solve problems involving exponents.

Conclusion

Simplifying exponential expressions can be challenging, but with practice and patience, you can become more confident in your ability to tackle these types of problems. Remember to apply the power of a power property, be careful when rewriting expressions with negative exponents as fractions, and check your work using a calculator or by plugging in values. If you're still having trouble, don't hesitate to ask for help.