Applying the Laws of Exponents

Oct 13, 2025 by miodeya.com 19 views

Simplify the Expression: $\left(9^4 \cdot 3^4\right)^{-1 / 4}$

Understanding the Problem

The given expression involves exponents and powers, and we are required to simplify it. To start, let's break down the expression and understand its components. The expression is $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ , which can be rewritten as $\left(3^8 \cdot 3^4\right)^{-1 / 4}$ .

Applying the Laws of Exponents

When we have the same base raised to different powers, we can add the exponents. In this case, we have $3^8$ and $3^4$ , which can be combined as $3^{8+4} = 3^{12}$ . Therefore, the expression becomes $\left(3^{12}\right)^{-1 / 4}$ .

Simplifying the Expression

Now, let's simplify the expression further. When we have a power raised to another power, we can multiply the exponents. In this case, we have $\left(3^{12}\right)^{-1 / 4}$ , which can be rewritten as $3^{12 \cdot (-1/4)}$ . This simplifies to $3^{-3}$ .

Evaluating the Expression

Now that we have simplified the expression to $3^{-3}$ , we can evaluate it. When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. In this case, $3^{-3}$ can be rewritten as $\frac{1}{3^3}$ .

Calculating the Final Value

Now, let's calculate the final value of the expression. We have $\frac{1}{3^3}$ , which can be evaluated as $\frac{1}{27}$ .

Conclusion

In this article, we simplified the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ by applying the laws of exponents and evaluating the resulting expression. We started by breaking down the expression and understanding its components, then applied the laws of exponents to simplify it further. Finally, we evaluated the expression and calculated the final value, which is $\frac{1}{27}$ .

Key Takeaways

When we have the same base raised to different powers, we can add the exponents.
When we have a power raised to another power, we can multiply the exponents.
When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base.

Real-World Applications

The laws of exponents have numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the laws of exponents are used to describe the behavior of particles and waves. In engineering, the laws of exponents are used to design and optimize systems. In economics, the laws of exponents are used to model and analyze economic systems.

Common Mistakes

When simplifying expressions involving exponents, it's common to make mistakes such as:

Forgetting to apply the laws of exponents
Not simplifying the expression fully
Not evaluating the expression correctly

Tips and Tricks

To simplify expressions involving exponents, follow these tips and tricks:

Break down the expression and understand its components
Apply the laws of exponents carefully
Simplify the expression fully
Evaluate the expression correctly

Conclusion

In conclusion, simplifying the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ requires careful application of the laws of exponents and evaluation of the resulting expression. By following the steps outlined in this article, you can simplify complex expressions involving exponents and arrive at the correct solution.
Simplify the Expression: $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ - Q&A

Q: What is the main concept behind simplifying the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ ?

A: The main concept behind simplifying the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ is to apply the laws of exponents and evaluate the resulting expression. This involves breaking down the expression, understanding its components, and simplifying it further.

Q: How do I simplify the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ ?

A: To simplify the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ , follow these steps:

Break down the expression and understand its components.
Apply the laws of exponents to simplify the expression.
Evaluate the resulting expression.

Q: What are the laws of exponents that I need to apply to simplify the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ ?

A: The laws of exponents that you need to apply to simplify the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ are:

When you have the same base raised to different powers, you can add the exponents.
When you have a power raised to another power, you can multiply the exponents.
When you have a negative exponent, you can rewrite it as a positive exponent by taking the reciprocal of the base.

Q: How do I apply the laws of exponents to simplify the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ ?

A: To apply the laws of exponents to simplify the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ , follow these steps:

Rewrite the expression as $\left(3^8 \cdot 3^4\right)^{-1 / 4}$ .
Combine the exponents using the law of exponents: $3^{8+4} = 3^{12}$ .
Rewrite the expression as $\left(3^{12}\right)^{-1 / 4}$ .
Multiply the exponents using the law of exponents: $3^{12 \cdot (-1/4)}$ .
Simplify the expression to $3^{-3}$ .

Q: How do I evaluate the expression $3^{-3}$ ?

A: To evaluate the expression $3^{-3}$ , take the reciprocal of the base and change the sign of the exponent. This gives you $\frac{1}{3^3}$ .

Q: What is the final value of the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ ?

A: The final value of the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ is $\frac{1}{27}$ .

Q: What are some common mistakes to avoid when simplifying expressions involving exponents?

A: Some common mistakes to avoid when simplifying expressions involving exponents are:

Forgetting to apply the laws of exponents
Not simplifying the expression fully
Not evaluating the expression correctly

Q: What are some tips and tricks for simplifying expressions involving exponents?

A: Some tips and tricks for simplifying expressions involving exponents are:

Break down the expression and understand its components
Apply the laws of exponents carefully
Simplify the expression fully
Evaluate the expression correctly

Conclusion

In this Q&A article, we have covered the main concepts and steps involved in simplifying the expression $\left(9^4 \cdot 3^4\right)^{-1 / 4}$ . We have also provided tips and tricks for simplifying expressions involving exponents and common mistakes to avoid. By following these steps and tips, you can simplify complex expressions involving exponents and arrive at the correct solution.