Which Expression Is Equivalent To ( A 2 / 3 B 1 / 3 ) 1 / 2 \left(a^{2 / 3} B^{1 / 3}\right)^{1 / 2} ( A 2/3 B 1/3 ) 1/2 ?A. A 7 / 6 B 5 / 6 A^{7 / 6} B^{5 / 6} A 7/6 B 5/6 B. A 4 / 3 B 2 / 3 A^{4 / 3} B^{2 / 3} A 4/3 B 2/3 C. A 1 / 2 B 1 / 2 A^{1 / 2} B^{1 / 2} A 1/2 B 1/2 D. A 4 / 3 B 1 / 6 A^{4 / 3} B^{1 / 6} A 4/3 B 1/6 E. $a^{1 / 3} B^{1 /

Mar 13, 2025 by ADMIN 347 views

**Simplifying Exponential Expressions: A Step-by-Step Guide**

===========================================================

Understanding Exponential Expressions

Exponential expressions are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. In this article, we will focus on simplifying exponential expressions, specifically the expression $\left(a^{2 / 3} b^{1 / 3}\right)^{1 / 2}$ .

The Power of a Product Rule

To simplify the given expression, we need to apply the power of a product rule, which states that for any non-zero numbers $a$ and $b$ , and any real numbers $m$ and $n$ ,

$\left(ab\right)^{m} = a^{m} b^{m}$

This rule allows us to distribute the exponent to each factor inside the parentheses.

Applying the Power of a Product Rule

Using the power of a product rule, we can rewrite the given expression as:

$\left(a^{2 / 3} b^{1 / 3}\right)^{1 / 2} = a^{(2 / 3) \cdot (1 / 2)} b^{(1 / 3) \cdot (1 / 2)}$

Simplifying the Exponents

Now, we can simplify the exponents by multiplying the fractions:

$a^{(2 / 3) \cdot (1 / 2)} = a^{1 / 3}$

$b^{(1 / 3) \cdot (1 / 2)} = b^{1 / 6}$

Combining the Simplified Exponents

Now that we have simplified the exponents, we can combine them to get the final expression:

$a^{1 / 3} b^{1 / 6}$

Comparing with the Answer Choices

Now, let's compare our final expression with the answer choices:

A. $a^{7 / 6} b^{5 / 6}$ B. $a^{4 / 3} b^{2 / 3}$ C. $a^{1 / 2} b^{1 / 2}$ D. $a^{4 / 3} b^{1 / 6}$ E. $a^{1 / 3} b^{1 / 6}$

Conclusion

Based on our simplification, we can see that the correct answer is:

E. $a^{1 / 3} b^{1 / 6}$

This is the equivalent expression to $\left(a^{2 / 3} b^{1 / 3}\right)^{1 / 2}$ .

Tips and Tricks

When simplifying exponential expressions, it's essential to apply the power of a product rule and simplify the exponents by multiplying the fractions. Additionally, make sure to combine the simplified exponents to get the final expression.

Common Mistakes to Avoid

When simplifying exponential expressions, some common mistakes to avoid include:

Not applying the power of a product rule
Not simplifying the exponents by multiplying the fractions
Not combining the simplified exponents to get the final expression

Real-World Applications

Exponential expressions have numerous real-world applications, including:

Modeling population growth and decay
Calculating compound interest
Analyzing electrical circuits

Practice Problems

To practice simplifying exponential expressions, try the following problems:

Simplify the expression $\left(a^{3 / 4} b^{2 / 4}\right)^{1 / 2}$
Simplify the expression $\left(a^{5 / 6} b^{3 / 6}\right)^{1 / 3}$

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill in mathematics, and it requires applying the power of a product rule and simplifying the exponents by multiplying the fractions. By following these steps and avoiding common mistakes, you can simplify exponential expressions with ease.

===========================================================

Q: What is the power of a product rule?

A: The power of a product rule states that for any non-zero numbers $a$ and $b$ , and any real numbers $m$ and $n$ ,

$\left(ab\right)^{m} = a^{m} b^{m}$

This rule allows us to distribute the exponent to each factor inside the parentheses.

Q: How do I apply the power of a product rule?

A: To apply the power of a product rule, simply distribute the exponent to each factor inside the parentheses. For example, if we have the expression $\left(ab\right)^{m}$ , we can rewrite it as $a^{m} b^{m}$ .

Q: What is the difference between a product and a power?

A: A product is the result of multiplying two or more numbers together, while a power is the result of raising a number to a certain exponent. For example, $a \cdot b$ is a product, while $a^{m}$ is a power.

Q: How do I simplify exponential expressions?

A: To simplify exponential expressions, follow these steps:

Apply the power of a product rule to distribute the exponent to each factor inside the parentheses.
Simplify the exponents by multiplying the fractions.
Combine the simplified exponents to get the final expression.

Q: What are some common mistakes to avoid when simplifying exponential expressions?

A: Some common mistakes to avoid when simplifying exponential expressions include:

Not applying the power of a product rule
Not simplifying the exponents by multiplying the fractions
Not combining the simplified exponents to get the final expression

Q: How do I know which answer choice is correct?

A: To determine which answer choice is correct, simply compare your final expression with the answer choices. If your final expression matches one of the answer choices, then that is the correct answer.

Q: What are some real-world applications of exponential expressions?

A: Exponential expressions have numerous real-world applications, including:

Modeling population growth and decay
Calculating compound interest
Analyzing electrical circuits

Q: Can you provide some practice problems to help me improve my skills?

A: Yes, here are some practice problems to help you improve your skills:

Simplify the expression $\left(a^{3 / 4} b^{2 / 4}\right)^{1 / 2}$
Simplify the expression $\left(a^{5 / 6} b^{3 / 6}\right)^{1 / 3}$

Q: How do I know if I'm doing the problem correctly?

A: To ensure that you're doing the problem correctly, follow these steps:

Read the problem carefully and understand what is being asked.
Apply the power of a product rule and simplify the exponents.
Combine the simplified exponents to get the final expression.
Compare your final expression with the answer choices to determine which one is correct.

Q: What if I'm still having trouble simplifying exponential expressions?

A: If you're still having trouble simplifying exponential expressions, try the following:

Review the power of a product rule and make sure you understand it.
Practice simplifying exponential expressions with different exponents and bases.
Ask a teacher or tutor for help if you're still struggling.