Simplify: ( 8 2 3 ) 4 \left(8^{\frac{2}{3}}\right)^4 ( 8 3 2 ​ ) 4 A. 8 8 3 8^{\frac{8}{3}} 8 3 8 ​ B. 8 6 3 8^{\frac{6}{3}} 8 3 6 ​ C. 8 6 12 8^{\frac{6}{12}} 8 12 6 ​

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Understanding Exponents and Power Rules

When dealing with exponents, it's essential to understand the rules that govern their behavior. One of the most critical rules is the power rule, which states that when a power is raised to another power, the exponents are multiplied. In mathematical terms, this can be represented as:

$(a^m)^n = a^{m \cdot n}$

This rule is crucial in simplifying expressions involving exponents.

Applying the Power Rule to the Given Expression

In the given expression, $\left(8^{\frac{2}{3}}\right)^4$, we can apply the power rule to simplify it. According to the rule, when a power is raised to another power, the exponents are multiplied. Therefore, we can rewrite the expression as:

$\left(8^{\frac{2}{3}}\right)^4 = 8^{\frac{2}{3} \cdot 4}$

Simplifying the Exponent

Now, we need to simplify the exponent $\frac{2}{3} \cdot 4$. To do this, we can multiply the numerator and denominator by 4:

$\frac{2}{3} \cdot 4 = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}$

Simplifying the Fraction

The fraction $\frac{8}{12}$ can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 4:

$\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$

Final Simplification

Now that we have simplified the exponent, we can rewrite the original expression as:

$\left(8^{\frac{2}{3}}\right)^4 = 8^{\frac{2}{3}}$

However, we can further simplify this expression by recognizing that $8^{\frac{2}{3}}$ is equivalent to $(8^2)^{\frac{1}{3}}$. Using the power rule again, we can rewrite this as:

$8^{\frac{2}{3}} = (8^2)^{\frac{1}{3}} = 8^{\frac{2}{3}}$

Conclusion

In conclusion, the simplified form of the given expression $\left(8^{\frac{2}{3}}\right)^4$ is $8^{\frac{8}{3}}$. This can be verified by applying the power rule and simplifying the resulting exponent.

Answer Options

A. $8^{\frac{8}{3}}$ B. $8^{\frac{6}{3}}$ C. $8^{\frac{6}{12}}$

Final Answer

The final answer is A. $8^{\frac{8}{3}}$.

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Understanding Exponents and Power Rules

When dealing with exponents, it's essential to understand the rules that govern their behavior. One of the most critical rules is the power rule, which states that when a power is raised to another power, the exponents are multiplied. In mathematical terms, this can be represented as:

$(a^m)^n = a^{m \cdot n}$

This rule is crucial in simplifying expressions involving exponents.

Q&A Session

Q: What is the power rule in exponents?

A: The power rule states that when a power is raised to another power, the exponents are multiplied. In mathematical terms, this can be represented as:

$(a^m)^n = a^{m \cdot n}$

Q: How do I apply the power rule to simplify expressions involving exponents?

A: To apply the power rule, you need to multiply the exponents. For example, if you have the expression $(a^m)^n$, you can rewrite it as $a^{m \cdot n}$.

Q: Can you give an example of how to apply the power rule?

A: Let's consider the expression $\left(8^{\frac{2}{3}}\right)^4$. To simplify this expression, we can apply the power rule by multiplying the exponents:

$\left(8^{\frac{2}{3}}\right)^4 = 8^{\frac{2}{3} \cdot 4}$

Q: How do I simplify the exponent $\frac{2}{3} \cdot 4$?

A: To simplify the exponent, you can multiply the numerator and denominator by 4:

$\frac{2}{3} \cdot 4 = \frac{2 \cdot 4}{3 \cdot 4} = \frac{8}{12}$

Q: Can you simplify the fraction $\frac{8}{12}$?

A: Yes, the fraction $\frac{8}{12}$ can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 4:

$\frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$

Q: What is the final simplified form of the expression $\left(8^{\frac{2}{3}}\right)^4$?

A: The final simplified form of the expression $\left(8^{\frac{2}{3}}\right)^4$ is $8^{\frac{8}{3}}$.

Common Mistakes to Avoid

• Not applying the power rule correctly
• Not simplifying the exponent
• Not recognizing that $8^{\frac{2}{3}}$ is equivalent to $(8^2)^{\frac{1}{3}}$

Tips for Simplifying Expressions Involving Exponents

• Always apply the power rule when simplifying expressions involving exponents
• Simplify the exponent by multiplying the numerator and denominator by the same value
• Recognize that $a^m$ is equivalent to $(a^n)^{\frac{m}{n}}$

Conclusion

In conclusion, the power rule is a crucial rule in simplifying expressions involving exponents. By applying the power rule and simplifying the resulting exponent, you can simplify complex expressions and arrive at the final answer.

Answer Options

A. $8^{\frac{8}{3}}$ B. $8^{\frac{6}{3}}$ C. $8^{\frac{6}{12}}$

Final Answer

The final answer is A. $8^{\frac{8}{3}}$.