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 ​

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Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is an essential 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 (x43x23)13\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, x2x^2 means xx multiplied by itself, or x⋅xx \cdot x. Similarly, x3x^3 means xx multiplied by itself three times, or x⋅x⋅xx \cdot x \cdot 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 any exponential expressions next.
  3. Multiplication and Division: Evaluate any 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 the given expression (x43x23)13\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 exponential expressions with the same base, we add the exponents. In this case, we have:

(x43x23)13=(x43+23)13\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}} = \left(x^{\frac{4}{3} + \frac{2}{3}}\right)^{\frac{1}{3}}

Using the product rule, we add the exponents:

43+23=63=2\frac{4}{3} + \frac{2}{3} = \frac{6}{3} = 2

So, the expression becomes:

(x2)13\left(x^2\right)^{\frac{1}{3}}

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:

(x2)13=x2⋅13\left(x^2\right)^{\frac{1}{3}} = x^{2 \cdot \frac{1}{3}}

Using the power rule, we multiply the exponents:

2⋅13=232 \cdot \frac{1}{3} = \frac{2}{3}

So, the expression becomes:

x23x^{\frac{2}{3}}

Conclusion

In conclusion, the expression (x43x23)13\left(x^{\frac{4}{3}} x^{\frac{2}{3}}\right)^{\frac{1}{3}} is equivalent to x23x^{\frac{2}{3}}. We applied the product rule to simplify the expression inside the parentheses and then used the power rule to simplify the resulting expression.

Answer

The correct answer is:

  • B. x23x^{\frac{2}{3}}

Final Thoughts

Q: What is the product rule for exponents?

A: The product rule states that when multiplying two exponential expressions with the same base, we add the exponents. For example, xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}.

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, (xa)b=xab(x^a)^b = x^{ab}.

Q: How do I simplify an exponential expression with a negative exponent?

A: To simplify an exponential expression with a negative exponent, we can rewrite it as a fraction with a positive exponent. For example, x−a=1xax^{-a} = \frac{1}{x^a}.

Q: What is the rule for dividing exponential expressions with the same base?

A: When dividing exponential expressions with the same base, we subtract the exponents. For example, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.

Q: How do I simplify an exponential expression with a variable in the exponent?

A: To simplify an exponential expression with a variable in the exponent, we can use the rules of exponents to combine the terms. For example, xa+b=xa⋅xbx^{a+b} = x^a \cdot x^b.

Q: What is the rule for raising an exponential expression to a power?

A: When raising an exponential expression to a power, we multiply the exponents. For example, (xa)b=xab(x^a)^b = x^{ab}.

Q: How do I simplify an exponential expression with a fraction in the exponent?

A: To simplify an exponential expression with a fraction in the exponent, we can use the rules of exponents to combine the terms. For example, xab=(xa)1bx^{\frac{a}{b}} = (x^a)^{\frac{1}{b}}.

Q: What is the rule for simplifying an exponential expression with a negative base?

A: When simplifying an exponential expression with a negative base, we can rewrite it as a positive base with a negative exponent. For example, (−x)a=(−1)a⋅xa(-x)^a = (-1)^a \cdot x^a.

Q: How do I simplify an exponential expression with a variable base and a variable exponent?

A: To simplify an exponential expression with a variable base and a variable exponent, we can use the rules of exponents to combine the terms. For example, xa+b=xa⋅xbx^{a+b} = x^a \cdot x^b.

Q: What is the rule for simplifying an exponential expression with a coefficient?

A: When simplifying an exponential expression with a coefficient, we can rewrite it as a product of the coefficient and the exponential expression. For example, abx=a⋅bxab^x = a \cdot b^x.

Conclusion

In conclusion, simplifying exponential expressions requires a solid understanding of the rules of exponents and the order of operations. By following these rules and applying them to a wide range of exponential expressions, we can simplify even the most complex expressions. Remember to always follow the order of operations and use the rules of exponents to combine terms.

Additional Resources

For more information on simplifying exponential expressions, check out the following resources:

  • Khan Academy: Exponents and Exponential Functions
  • Mathway: Exponents and Exponential Functions
  • Wolfram Alpha: Exponents and Exponential Functions

Final Thoughts

Simplifying exponential expressions is an essential skill for students and professionals alike. By mastering the rules of exponents and the order of operations, we can simplify even the most complex expressions and solve a wide range of mathematical problems. Remember to always follow the order of operations and use the rules of exponents to combine terms.