Which Expression Is Equivalent To ( X 4 3 X 2 3 ) 1 3 \left(x^{\frac{4}{3}} X^{\frac{2}{3}}\right)^{\frac{1}{3}} ( X 3 4 ​ X 3 2 ​ ) 3 1 ​ ?A. X 2 9 X^{\frac{2}{9}} X 9 2 ​ B. X 2 3 X^{\frac{2}{3}} X 3 2 ​ C. X 8 27 X^{\frac{8}{27}} X 27 8 ​ D. X 7 3 X^{\frac{7}{3}} X 3 7 ​

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying exponential expressions, with a focus on the given expression $\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}$. We will break down the expression into manageable parts, apply the rules of exponents, and arrive at the equivalent expression.

Understanding Exponents

Before we dive into the simplification process, let's review the basics of exponents. An exponent is a small number that is placed above and to the right of a base number. It represents the number of times the base number is multiplied by itself. For example, $x^2$ means $x$ multiplied by itself, or $x \times x$. Similarly, $x^3$ means $x$ multiplied by itself three times, or $x \times x \times x$.

The Order of Operations

When simplifying exponential expressions, it's essential to follow the order of operations (PEMDAS):

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

Simplifying the Given Expression

Now that we've reviewed the basics of exponents and the order of operations, let's apply these concepts to simplify the given expression $\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}$.

Step 1: Apply the Product Rule

The product rule states that when multiplying two or more exponential expressions with the same base, we add the exponents. In this case, we have $x^{\frac{4}{3}}$ and $x^{\frac{2}{3}}$, both with base $x$. Applying the product rule, we get:

$$\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right) = x^{\frac{4}{3} + \frac{2}{3}} = x^{\frac{6}{3}} = x^2$$

Step 2: Apply the Power Rule

The power rule states that when raising an exponential expression to a power, we multiply the exponents. In this case, we have $x^2$ raised to the power of $\frac{1}{3}$. Applying the power rule, we get:

$$\left(x^2\right)^{\frac{1}{3}} = x^{2 \times \frac{1}{3}} = x^{\frac{2}{3}}$$

Conclusion

In conclusion, the expression $\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}$ is equivalent to $x^{\frac{2}{3}}$. This result can be verified by applying the product rule and the power rule, as demonstrated in the previous steps.

Answer

The correct answer is:

• B. $x^{\frac{2}{3}}$

Final Thoughts

Q: What is the product rule for exponents?

A: The product rule states that when multiplying two or more exponential expressions with the same base, we add the exponents. For example, $x^2 \times x^3 = x^{2+3} = x^5$.

Q: What is the power rule for exponents?

A: The power rule states that when raising an exponential expression to a power, we multiply the exponents. For example, $(x^2)^3 = x^{2 \times 3} = x^6$.

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

A: To simplify an expression with multiple exponents, apply the product rule and the power rule in the following order:

1. Evaluate any expressions inside parentheses.
2. Apply the product rule to combine any exponential expressions with the same base.
3. Apply the power rule to raise any exponential expressions to a power.

Q: What is the order of operations for simplifying exponential expressions?

A: The order of operations for simplifying exponential expressions is:

1. Evaluate any expressions inside parentheses.
2. Apply the product rule to combine any exponential expressions with the same base.
3. Apply the power rule to raise any exponential expressions to a power.
4. Evaluate any multiplication and division operations from left to right.
5. Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I handle negative exponents?

A: When simplifying an expression with a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, $x^{-2} = \frac{1}{x^2}$.

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent by applying the product rule and the power rule. For example, $(x^2)^3 = x^{2 \times 3} = x^6$.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is an expression that contains a base raised to a power, such as $x^2$. A polynomial expression is an expression that contains variables and coefficients, such as $2x^2 + 3x - 1$.

Q: Can I simplify an expression with multiple bases?

A: Yes, you can simplify an expression with multiple bases by applying the product rule and the power rule. For example, $x^2 \times y^3 = (x \times y)^{2+3} = (xy)^5$.

Q: How do I simplify an expression with a fraction as an exponent?

A: To simplify an expression with a fraction as an exponent, apply the power rule and the product rule in the following order:

1. Evaluate any expressions inside parentheses.
2. Apply the power rule to raise any exponential expressions to a power.
3. Apply the product rule to combine any exponential expressions with the same base.

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent by applying the rule that any number raised to the power of zero is equal to 1. For example, $x^0 = 1$.

Q: What is the final answer to the original expression $\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}}$?

A: The final answer to the original expression is $x^{\frac{2}{3}}$.