What Is The Value Of The Expression Below? 64 4 / 6 64^{4/6} 6 4 4/6 A. 42 B. 16 C. 2 D. 8

Understanding the Expression

The given expression is $64^{4/6}$. To evaluate this expression, we need to understand the properties of exponents and fractions. The expression involves a base of 64 and an exponent of 4/6.

Simplifying the Fractional Exponent

To simplify the expression, we can start by simplifying the fractional exponent. The exponent 4/6 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

4/6 = 2/3

Applying the Simplified Exponent

Now that we have simplified the exponent, we can rewrite the expression as $64^{2/3}$.

Understanding the Base

The base of the expression is 64, which can be expressed as a power of 2. Specifically, 64 is equal to $2^6$.

64 = 2^6

Applying the Base to the Exponent

Now that we have expressed the base as a power of 2, we can rewrite the expression as $(2^6)^{2/3}$.

Applying the Power of a Power Rule

The expression $(2^6)^{2/3}$ can be simplified using the power of a power rule, which states that $(a^m)^n = a^{mn}$.

(2^6)^{2/3} = 2^{6 \cdot (2/3)}

Simplifying the Exponent

The exponent $6 \cdot (2/3)$ can be simplified by multiplying 6 by 2 and then dividing by 3.

6 \cdot (2/3) = 4

Applying the Simplified Exponent

Now that we have simplified the exponent, we can rewrite the expression as $2^4$.

Evaluating the Expression

The expression $2^4$ can be evaluated by multiplying 2 by itself 4 times.

2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16

Conclusion

The value of the expression $64^{4/6}$ is 16.

Final Answer

The final answer is $\boxed{16}$.

Discussion

The expression $64^{4/6}$ can be evaluated by simplifying the fractional exponent, expressing the base as a power of 2, and applying the power of a power rule. The final answer is 16, which is the result of evaluating the expression $2^4$.

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Q: What is the value of the expression $64^{4/6}$?

A: The value of the expression $64^{4/6}$ is 16.

Q: How do I simplify the fractional exponent 4/6?

A: To simplify the fractional exponent 4/6, you can divide both the numerator and the denominator by their greatest common divisor, which is 2. This results in a simplified exponent of 2/3.

Q: What is the base of the expression $64^{4/6}$?

A: The base of the expression $64^{4/6}$ is 64, which can be expressed as a power of 2. Specifically, 64 is equal to $2^6$.

Q: How do I apply the power of a power rule to the expression $(2^6)^{2/3}$?

A: To apply the power of a power rule, you can multiply the exponent of the inner power by the exponent of the outer power. In this case, the exponent of the inner power is 6 and the exponent of the outer power is 2/3. This results in a simplified exponent of 4.

Q: What is the value of the expression $2^4$?

A: The value of the expression $2^4$ is 16, which is the result of multiplying 2 by itself 4 times.

Q: Why is the value of the expression $64^{4/6}$ equal to 16?

A: The value of the expression $64^{4/6}$ is equal to 16 because it can be simplified to $2^4$, which is equal to 16.

Q: Can I use a calculator to evaluate the expression $64^{4/6}$?

A: Yes, you can use a calculator to evaluate the expression $64^{4/6}$. However, it's also important to understand the underlying math and be able to simplify the expression manually.

Q: What are some common mistakes to avoid when evaluating the expression $64^{4/6}$?

A: Some common mistakes to avoid when evaluating the expression $64^{4/6}$ include:

• Not simplifying the fractional exponent
• Not expressing the base as a power of 2
• Not applying the power of a power rule correctly
• Not evaluating the expression correctly

Q: How can I practice evaluating expressions like $64^{4/6}$?

A: You can practice evaluating expressions like $64^{4/6}$ by working through a series of practice problems. You can also try simplifying different expressions and evaluating them to see if you get the correct answer.

Q: What are some real-world applications of evaluating expressions like $64^{4/6}$?

A: Evaluating expressions like $64^{4/6}$ has many real-world applications, including:

• Calculating interest rates and investments
• Determining the size of a population or sample
• Evaluating the effectiveness of a marketing campaign
• Calculating the cost of a project or product

Q: Can I use the expression $64^{4/6}$ in a real-world scenario?

A: Yes, you can use the expression $64^{4/6}$ in a real-world scenario. For example, you might use it to calculate the size of a population or sample, or to determine the effectiveness of a marketing campaign.

Q: What are some tips for evaluating expressions like $64^{4/6}$ in a real-world scenario?

A: Some tips for evaluating expressions like $64^{4/6}$ in a real-world scenario include:

• Make sure to simplify the expression correctly
• Use the correct order of operations
• Check your work to make sure you get the correct answer
• Consider using a calculator or computer program to evaluate the expression if necessary

Q: Can I use the expression $64^{4/6}$ in a mathematical proof or theorem?

A: Yes, you can use the expression $64^{4/6}$ in a mathematical proof or theorem. For example, you might use it to prove a theorem about exponents or to demonstrate a mathematical concept.

Q: What are some common mathematical concepts that involve evaluating expressions like $64^{4/6}$?

A: Some common mathematical concepts that involve evaluating expressions like $64^{4/6}$ include:

• Exponents and powers
• Fractions and decimals
• Algebra and equations
• Geometry and trigonometry

Q: Can I use the expression $64^{4/6}$ in a scientific or engineering application?

A: Yes, you can use the expression $64^{4/6}$ in a scientific or engineering application. For example, you might use it to calculate the size of a population or sample, or to determine the effectiveness of a scientific experiment.

Q: What are some tips for evaluating expressions like $64^{4/6}$ in a scientific or engineering application?

A: Some tips for evaluating expressions like $64^{4/6}$ in a scientific or engineering application include:

• Make sure to simplify the expression correctly
• Use the correct order of operations
• Check your work to make sure you get the correct answer
• Consider using a calculator or computer program to evaluate the expression if necessary

Q: Can I use the expression $64^{4/6}$ in a programming or coding application?

A: Yes, you can use the expression $64^{4/6}$ in a programming or coding application. For example, you might use it to calculate the size of a population or sample, or to determine the effectiveness of a program or algorithm.

Q: What are some tips for evaluating expressions like $64^{4/6}$ in a programming or coding application?

A: Some tips for evaluating expressions like $64^{4/6}$ in a programming or coding application include:

• Make sure to simplify the expression correctly
• Use the correct order of operations
• Check your work to make sure you get the correct answer
• Consider using a calculator or computer program to evaluate the expression if necessary