Which Is The Simplified Form Of $n^{-6} P^3$?A. $\frac{n^6}{p^3}$B. $\frac{1}{n^6 P^3}$C. $\frac{p^3}{n^6}$D. $n^6 P^3$

Understanding Exponents

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the simplified form of the expression $n^{-6} p^3$.

What are Exponents?

Exponents are a shorthand way of writing repeated multiplication. For example, $n^3$ can be read as "n to the power of 3" and is equivalent to $n \times n \times n$. Exponents can be positive, negative, or zero.

Negative Exponents

A negative exponent is a fraction with the exponent in the denominator. For example, $n^{-3}$ is equivalent to $\frac{1}{n^3}$. This is a crucial concept in simplifying expressions with exponents.

Simplifying the Expression

Now, let's simplify the expression $n^{-6} p^3$. To do this, we need to apply the rules of exponents.

Rule 1: Negative Exponents

When a negative exponent is present, we can rewrite it as a fraction with the exponent in the denominator. In this case, $n^{-6}$ can be rewritten as $\frac{1}{n^6}$.

Rule 2: Multiplying Exponents

When multiplying two or more numbers with exponents, we add the exponents. In this case, we have $p^3$ and $\frac{1}{n^6}$. To multiply these two expressions, we add the exponents: $p^3 \times \frac{1}{n^6} = \frac{p^3}{n^6}$.

Simplified Form

Now that we have applied the rules of exponents, we can simplify the expression $n^{-6} p^3$.

The Correct Answer

The simplified form of $n^{-6} p^3$ is $\frac{p^3}{n^6}$.

Conclusion

In this article, we have explored the simplified form of the expression $n^{-6} p^3$. We have applied the rules of exponents, including negative exponents and multiplying exponents. The correct answer is $\frac{p^3}{n^6}$.

Common Mistakes

When simplifying expressions with exponents, it's essential to remember the rules of exponents. Some common mistakes include:

• Not applying the rules of exponents: Failing to apply the rules of exponents can lead to incorrect simplifications.
• Incorrectly handling negative exponents: Negative exponents can be tricky to handle, but it's essential to remember that they can be rewritten as fractions with the exponent in the denominator.
• Not considering the order of operations: When simplifying expressions with exponents, it's essential to consider the order of operations (PEMDAS).

Practice Problems

To reinforce your understanding of simplifying expressions with exponents, try the following practice problems:

• Simplify the expression $2^{-3} x^4$.
• Simplify the expression $3^2 y^{-5}$.
• Simplify the expression $4^{-2} z^3$.

Answer Key

• $2^{-3} x^4 = \frac{x^4}{2^3} = \frac{x^4}{8}$
• $3^2 y^{-5} = \frac{1}{3^2 y^5} = \frac{1}{9y^5}$
• $4^{-2} z^3 = \frac{z^3}{4^2} = \frac{z^3}{16}$

Conclusion

Q: What is the simplified form of $n^{-3} p^2$?

A: To simplify the expression $n^{-3} p^2$, we need to apply the rules of exponents. We can rewrite $n^{-3}$ as $\frac{1}{n^3}$ and then multiply it by $p^2$. This gives us $\frac{p^2}{n^3}$.

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

A: To simplify an expression with a negative exponent and a positive exponent, we need to apply the rules of exponents. We can rewrite the negative exponent as a fraction with the exponent in the denominator and then multiply it by the positive exponent. For example, to simplify $n^{-2} p^3$, we can rewrite $n^{-2}$ as $\frac{1}{n^2}$ and then multiply it by $p^3$. This gives us $\frac{p^3}{n^2}$.

Q: What is the simplified form of $2^{-4} 3^2$?

A: To simplify the expression $2^{-4} 3^2$, we need to apply the rules of exponents. We can rewrite $2^{-4}$ as $\frac{1}{2^4}$ and then multiply it by $3^2$. This gives us $\frac{3^2}{2^4} = \frac{9}{16}$.

Q: How do I simplify an expression with multiple negative exponents?

A: To simplify an expression with multiple negative exponents, we need to apply the rules of exponents. We can rewrite each negative exponent as a fraction with the exponent in the denominator and then multiply the fractions together. For example, to simplify $n^{-2} p^{-3}$, we can rewrite $n^{-2}$ as $\frac{1}{n^2}$ and $p^{-3}$ as $\frac{1}{p^3}$. Then, we can multiply the fractions together to get $\frac{1}{n^2 p^3}$.

Q: What is the simplified form of $x^{-2} y^{-3}$?

A: To simplify the expression $x^{-2} y^{-3}$, we need to apply the rules of exponents. We can rewrite $x^{-2}$ as $\frac{1}{x^2}$ and $y^{-3}$ as $\frac{1}{y^3}$. Then, we can multiply the fractions together to get $\frac{1}{x^2 y^3}$.

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

A: To simplify an expression with a negative exponent and a fraction, we need to apply the rules of exponents. We can rewrite the negative exponent as a fraction with the exponent in the denominator and then multiply it by the fraction. For example, to simplify $\frac{1}{2^{-3}}$, we can rewrite $2^{-3}$ as $\frac{1}{2^3}$ and then multiply it by $\frac{1}{1}$. This gives us $\frac{1}{\frac{1}{2^3}} = 2^3 = 8$.

Q: What is the simplified form of $\frac{1}{2^{-2}}$?

A: To simplify the expression $\frac{1}{2^{-2}}$, we can rewrite $2^{-2}$ as $\frac{1}{2^2}$ and then multiply it by $\frac{1}{1}$. This gives us $\frac{1}{\frac{1}{2^2}} = 2^2 = 4$.

Conclusion

In this article, we have answered some frequently asked questions about simplifying exponents. We have applied the rules of exponents, including negative exponents and multiplying exponents. Remember to practice and reinforce your understanding of simplifying expressions with exponents.