What Is The Value Of The Expression Below?$\left(64^3\right)^{1 / 6}$A. 4 B. 8 C. 32 D. 64

Understanding the Expression

The given expression is $\left(64^3\right)^{1 / 6}$. To evaluate this expression, we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this case, we have an exponent inside the parentheses, and another exponent outside the parentheses.

Simplifying the Expression

To simplify the expression, we can start by evaluating the exponent inside the parentheses. We have $64^3$, which means $64$ multiplied by itself $3$ times. This can be written as $64 \times 64 \times 64$.

Evaluating the Exponent Inside the Parentheses

$64 \times 64 \times 64 = 262,144$. Now, we have $\left(262,144\right)^{1 / 6}$.

Evaluating the Exponent Outside the Parentheses

To evaluate the exponent outside the parentheses, we can use the rule of exponents, which states that $\left(a^m\right)^n = a^{m \times n}$. In this case, we have $\left(262,144\right)^{1 / 6}$, which can be written as $262,144^{1 / 6}$.

Simplifying the Expression Further

To simplify the expression further, we can use the rule of exponents again. We have $262,144^{1 / 6}$, which can be written as $\left(262,144\right)^{1/6}$. Using the rule of exponents, we can rewrite this as $262,144^{1/6} = \left(262,144\right)^{1/6}$.

Evaluating the Final Expression

To evaluate the final expression, we can use the fact that $a^{1/n} = \sqrt[n]{a}$. In this case, we have $\left(262,144\right)^{1/6}$, which can be written as $\sqrt[6]{262,144}$.

Simplifying the Final Expression

To simplify the final expression, we can use the fact that $\sqrt[n]{a} = a^{1/n}$. In this case, we have $\sqrt[6]{262,144}$, which can be written as $262,144^{1/6}$.

Evaluating the Final Answer

To evaluate the final answer, we can use the fact that $262,144 = 4^6$. Therefore, we have $262,144^{1/6} = \left(4^6\right)^{1/6}$.

Simplifying the Final Answer

To simplify the final answer, we can use the rule of exponents again. We have $\left(4^6\right)^{1/6}$, which can be written as $4^{6 \times 1/6}$.

Evaluating the Final Answer

To evaluate the final answer, we can simplify the exponent. We have $4^{6 \times 1/6}$, which can be written as $4^1$.

The Final Answer

The final answer is $\boxed{4}$.

Conclusion

In this article, we have evaluated the expression $\left(64^3\right)^{1 / 6}$ and found that the final answer is $\boxed{4}$. This is a great example of how to use the order of operations and the rule of exponents to simplify complex expressions.

Frequently Asked Questions

• Q: What is the value of the expression $\left(64^3\right)^{1 / 6}$?
• A: The value of the expression is $\boxed{4}$.
• Q: How do I evaluate the expression $\left(64^3\right)^{1 / 6}$?
• A: To evaluate the expression, you can use the order of operations and the rule of exponents.
• Q: What is the rule of exponents?
• A: The rule of exponents states that $\left(a^m\right)^n = a^{m \times n}$.

Related Articles

• Evaluating Expressions with Exponents
• The Order of Operations
• Simplifying Complex Expressions
  

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate an expression with exponents?

A: To evaluate an expression with exponents, you need to follow the order of operations. First, evaluate any expressions inside parentheses. Next, evaluate any exponential expressions. Finally, evaluate any multiplication and division operations from left to right, and then any addition and subtraction operations from left to right.

Q: What is the rule of exponents?

A: The rule of exponents states that $\left(a^m\right)^n = a^{m \times n}$. This means that when we have an exponential expression inside another exponential expression, we can simplify it by multiplying the exponents.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to follow the rule of exponents. If you have an expression like $\left(a^m\right)^n$, you can simplify it by multiplying the exponents, like this: $\left(a^m\right)^n = a^{m \times n}$.

Q: What is the difference between $a^m$ and $a^{m/n}$?

A: $a^m$ means that $a$ is multiplied by itself $m$ times, while $a^{m/n}$ means that $a$ is multiplied by itself $m$ times, and then the result is divided by $n$.

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

A: To evaluate an expression with a negative exponent, you need to follow the rule of exponents. If you have an expression like $a^{-m}$, you can simplify it by taking the reciprocal of $a^m$, like this: $a^{-m} = \frac{1}{a^m}$.

Q: What is the value of the expression $\left(64^3\right)^{1 / 6}$?

A: The value of the expression $\left(64^3\right)^{1 / 6}$ is $4$.

Q: How do I evaluate an expression with a fractional exponent?

A: To evaluate an expression with a fractional exponent, you need to follow the rule of exponents. If you have an expression like $a^{m/n}$, you can simplify it by taking the $n$th root of $a^m$, like this: $a^{m/n} = \sqrt[n]{a^m}$.

Q: What is the value of the expression $\left(64^3\right)^{1 / 6}$?

A: The value of the expression $\left(64^3\right)^{1 / 6}$ is $4$.

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

A: To simplify an expression with multiple exponents, you need to follow the rule of exponents. If you have an expression like $\left(a^m\right)^n$, you can simplify it by multiplying the exponents, like this: $\left(a^m\right)^n = a^{m \times n}$.

Q: What is the value of the expression $\left(64^3\right)^{1 / 6}$?

A: The value of the expression $\left(64^3\right)^{1 / 6}$ is $4$.

Q: How do I evaluate an expression with a variable exponent?

A: To evaluate an expression with a variable exponent, you need to follow the rule of exponents. If you have an expression like $a^{m/n}$, you can simplify it by taking the $n$th root of $a^m$, like this: $a^{m/n} = \sqrt[n]{a^m}$.

Q: What is the value of the expression $\left(64^3\right)^{1 / 6}$?

A: The value of the expression $\left(64^3\right)^{1 / 6}$ is $4$.

Conclusion

In this article, we have answered some frequently asked questions about evaluating expressions with exponents. We have covered topics such as the order of operations, the rule of exponents, and how to simplify expressions with multiple exponents. We have also provided examples of how to evaluate expressions with negative exponents, fractional exponents, and variable exponents.

Related Articles

• Evaluating Expressions with Exponents
• The Order of Operations
• Simplifying Complex Expressions

Further Reading

• Algebraic Expressions
• Exponents and Powers
• Mathematical Operations