What Is The Value Of The Expression Below? ( 81 2 ) 1 / 6 \left(81^2\right)^{1 / 6} ( 8 1 2 ) 1/6 A. 9 B. 45 C. 405 D. 81

Understanding the Problem

The given expression is $\left(81^2\right)^{1 / 6}$. To find the value of this expression, we need to simplify it step by step. The expression involves exponentiation and fractional exponents, so we will use the properties of exponents to simplify it.

Step 1: Simplify the Inner Exponent

The inner exponent is $81^2$. We can simplify this by calculating the value of $81^2$.

$81^2 = 81 \times 81 = 6561$

So, the expression becomes $\left(6561\right)^{1 / 6}$.

Step 2: Simplify the Outer Exponent

The outer exponent is $1/6$. We can simplify this by raising the inner expression to the power of $1/6$.

$\left(6561\right)^{1 / 6} = \sqrt[6]{6561}$

Step 3: Evaluate the Expression

Now, we need to evaluate the expression $\sqrt[6]{6561}$. To do this, we can use the fact that $6561 = 3^8$. So, we can rewrite the expression as:

$\sqrt[6]{6561} = \sqrt[6]{3^8}$

Using the property of exponents that $\sqrt[n]{a^n} = a$, we can simplify this to:

$\sqrt[6]{3^8} = 3^{8/6}$

Step 4: Simplify the Fractional Exponent

The fractional exponent $8/6$ can be simplified by dividing the numerator and denominator by their greatest common divisor, which is 2.

$8/6 = 4/3$

So, the expression becomes:

$3^{4/3}$

Step 5: Evaluate the Expression

Now, we need to evaluate the expression $3^{4/3}$. To do this, we can use the fact that $3^{4/3} = (3^4)^{1/3}$.

$3^4 = 3 \times 3 \times 3 \times 3 = 81$

So, the expression becomes:

$(81)^{1/3}$

Step 6: Evaluate the Expression

Finally, we need to evaluate the expression $(81)^{1/3}$. To do this, we can use the fact that $81 = 3^4$. So, we can rewrite the expression as:

$(81)^{1/3} = (3^4)^{1/3}$

Using the property of exponents that $(a^m)^n = a^{mn}$, we can simplify this to:

$3^{4/3}$

Conclusion

The value of the expression $\left(81^2\right)^{1 / 6}$ is $3^{4/3}$. This can be evaluated as $(81)^{1/3}$, which is equal to $\boxed{9}$.

Answer

Frequently Asked Questions

We have received several questions from readers regarding the value of the expression $\left(81^2\right)^{1 / 6}$. Here are some of the most frequently asked questions and their answers:

Q: What is the value of the expression $\left(81^2\right)^{1 / 6}$?

A: The value of the expression $\left(81^2\right)^{1 / 6}$ is $3^{4/3}$, which can be evaluated as $(81)^{1/3}$.

Q: How do I simplify the expression $\left(81^2\right)^{1 / 6}$?

A: To simplify the expression $\left(81^2\right)^{1 / 6}$, you can follow these steps:

1. Simplify the inner exponent $81^2$ to get $6561$.
2. Raise the inner expression to the power of $1/6$ to get $\sqrt[6]{6561}$.
3. Use the fact that $6561 = 3^8$ to rewrite the expression as $\sqrt[6]{3^8}$.
4. Simplify the fractional exponent $8/6$ to get $4/3$.
5. Evaluate the expression $3^{4/3}$ to get $(81)^{1/3}$.

Q: What is the relationship between the expression $\left(81^2\right)^{1 / 6}$ and the number $9$?

A: The expression $\left(81^2\right)^{1 / 6}$ is equal to $9$. This can be seen by evaluating the expression $(81)^{1/3}$, which is equal to $9$.

Q: Can I use a calculator to evaluate the expression $\left(81^2\right)^{1 / 6}$?

A: Yes, you can use a calculator to evaluate the expression $\left(81^2\right)^{1 / 6}$. However, it is recommended to follow the steps outlined above to understand the underlying mathematics.

Q: What is the significance of the expression $\left(81^2\right)^{1 / 6}$ in mathematics?

A: The expression $\left(81^2\right)^{1 / 6}$ is a simple example of how to work with exponents and fractional exponents. It is a fundamental concept in mathematics and is used in a variety of applications, including algebra, geometry, and calculus.

Additional Resources

For more information on exponents and fractional exponents, please refer to the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram MathWorld: Exponents and Exponential Functions

Conclusion

We hope this Q&A article has helped to clarify any questions you may have had regarding the value of the expression $\left(81^2\right)^{1 / 6}$. If you have any further questions or concerns, please don't hesitate to contact us.