Simplify:$\[ \left((x-5)^{\frac{3}{2}}\right)^{\frac{2}{3}} \\]Options:A. \[$x-5\$\] B. \[$(x-5)^2\$\] C. \[$(x-5)^3\$\]

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Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. In this problem, we're given the expression $\left((x-5)^{\frac{3}{2}}\right)^{\frac{2}{3}}$. Our goal is to simplify this expression and choose the correct option from the given choices.

Applying the Power Rule of Exponents

To simplify the given expression, we can use the power rule of exponents, which states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. In this case, we have $\left((x-5)^{\frac{3}{2}}\right)^{\frac{2}{3}}$, so we can apply the power rule by multiplying the exponents.

Simplifying the Expression

Using the power rule, we can simplify the expression as follows:

$\left((x-5)^{\frac{3}{2}}\right)^{\frac{2}{3}} = (x-5)^{\frac{3}{2} \cdot \frac{2}{3}}$

Evaluating the Exponents

Now, we need to evaluate the product of the exponents. Multiplying the numerators and denominators, we get:

$\frac{3}{2} \cdot \frac{2}{3} = \frac{6}{6} = 1$

Simplifying the Expression Further

Since the product of the exponents is 1, we can simplify the expression further by removing the exponent:

$(x-5)^1 = x-5$

Choosing the Correct Option

Now that we have simplified the expression, we can choose the correct option from the given choices. The simplified expression is $x-5$, which matches option A.

Conclusion

In this problem, we applied the power rule of exponents to simplify the given expression. We evaluated the product of the exponents and removed the exponent to get the final simplified expression. The correct option is A. $\boxed{x-5}$

Discussion

This problem requires a good understanding of the power rule of exponents and how to apply it to simplify expressions. It's essential to remember that when dealing with exponents, the order of operations matters, and we need to follow the rules of exponentiation carefully.

Additional Examples

Here are a few additional examples to help reinforce the concept:

• $\left((x+2)^{\frac{4}{3}}\right)^{\frac{3}{4}} = (x+2)^{\frac{4}{3} \cdot \frac{3}{4}} = (x+2)^1 = x+2$
• $\left((x-1)^{\frac{5}{2}}\right)^{\frac{2}{5}} = (x-1)^{\frac{5}{2} \cdot \frac{2}{5}} = (x-1)^1 = x-1$
• $\left((x+3)^{\frac{3}{4}}\right)^{\frac{4}{3}} = (x+3)^{\frac{3}{4} \cdot \frac{4}{3}} = (x+3)^1 = x+3$

These examples demonstrate how to apply the power rule of exponents to simplify expressions and choose the correct option.

Final Thoughts

In conclusion, simplifying expressions with exponents requires a good understanding of the power rule of exponents and how to apply it carefully. By following the rules of exponentiation and evaluating the product of the exponents, we can simplify expressions and choose the correct option.

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Understanding the Problem

When dealing with exponents, it's essential to remember the rules of exponentiation. In this problem, we're given the expression $\left((x-5)^{\frac{3}{2}}\right)^{\frac{2}{3}}$. Our goal is to simplify this expression and choose the correct option from the given choices.

Q&A

Q: What is the power rule of exponents?

A: The power rule of exponents states that for any numbers $a$ and $b$ and any integers $m$ and $n$, $(a^m)^n = a^{m \cdot n}$. This rule allows us to simplify expressions with exponents by multiplying the exponents.

Q: How do I apply the power rule of exponents?

A: To apply the power rule, we need to multiply the exponents. For example, if we have $(x-5)^{\frac{3}{2}}$, we can apply the power rule by multiplying the exponents: $\left((x-5)^{\frac{3}{2}}\right)^{\frac{2}{3}} = (x-5)^{\frac{3}{2} \cdot \frac{2}{3}}$.

Q: What happens when I multiply the exponents?

A: When we multiply the exponents, we need to evaluate the product. In this case, we have $\frac{3}{2} \cdot \frac{2}{3} = \frac{6}{6} = 1$. This means that the exponent is simplified to 1.

Q: How do I simplify the expression further?

A: Since the exponent is 1, we can simplify the expression further by removing the exponent: $(x-5)^1 = x-5$.

Q: What is the correct option?

A: The correct option is A. $\boxed{x-5}$.

Common Mistakes

Mistake 1: Not applying the power rule of exponents

• Not applying the power rule of exponents can lead to incorrect simplification of expressions.
• Make sure to apply the power rule by multiplying the exponents.

Mistake 2: Not evaluating the product of the exponents

• Not evaluating the product of the exponents can lead to incorrect simplification of expressions.
• Make sure to evaluate the product of the exponents carefully.

Mistake 3: Not simplifying the expression further

• Not simplifying the expression further can lead to incorrect answers.
• Make sure to simplify the expression further by removing the exponent.

Additional Tips

Tip 1: Practice, practice, practice

• Practice simplifying expressions with exponents to become more comfortable with the power rule.
• Start with simple expressions and gradually move on to more complex ones.

Tip 2: Use visual aids

• Use visual aids such as diagrams or charts to help you understand the power rule of exponents.
• Visual aids can help you see the relationships between the exponents and the expression.

Tip 3: Check your work

• Check your work carefully to ensure that you have simplified the expression correctly.
• Make sure to evaluate the product of the exponents and simplify the expression further.

Conclusion

In conclusion, simplifying expressions with exponents requires a good understanding of the power rule of exponents and how to apply it carefully. By following the rules of exponentiation and evaluating the product of the exponents, we can simplify expressions and choose the correct option. Remember to practice, use visual aids, and check your work to become more comfortable with the power rule of exponents.