Which Expression Is Equivalent To $x^{-\frac{5}{3}}$?A. $\frac{1}{\sqrt[5]{x^3}}$B. \$\frac{1}{\sqrt[3]{x^5}}$[/tex\]C. $-\sqrt[3]{x^5}$D. $-\sqrt[5]{x^3}$

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Introduction

Exponential expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $x^{-\frac{5}{3}}$ and explore its equivalent forms. We will examine four different options and determine which one is equivalent to the given expression.

Understanding Negative Exponents

Before we dive into the simplification process, it's essential to understand the concept of negative exponents. A negative exponent indicates that the reciprocal of the base is raised to the power of the exponent. In other words, $x^{-a} = \frac{1}{x^a}$. This concept is crucial for simplifying exponential expressions.

Option A: $\frac{1}{\sqrt[5]{x^3}}$

Let's start by analyzing Option A: $\frac1}{\sqrt[5]{x^3}}$. To simplify this expression, we need to rewrite it in a more familiar form. We can start by expressing the square root as a fractional exponent $\sqrt[5]{x^3 = x^\frac{3}{5}}$. Now, we can rewrite the expression as $\frac{1{x^{\frac{3}{5}}} = x^{-\frac{3}{5}}$. However, this is not equivalent to the original expression $x^{-\frac{5}{3}}$. Therefore, Option A is not the correct answer.

Option B: $\frac{1}{\sqrt[3]{x^5}}$

Next, let's analyze Option B: $\frac1}{\sqrt[3]{x^5}}$. Similar to Option A, we can rewrite the square root as a fractional exponent $\sqrt[3]{x^5 = x^\frac{5}{3}}$. Now, we can rewrite the expression as $\frac{1{x^{\frac{5}{3}}} = x^{-\frac{5}{3}}$. This is equivalent to the original expression $x^{-\frac{5}{3}}$. Therefore, Option B is the correct answer.

Option C: $-\sqrt[3]{x^5}$

Option C: $-\sqrt[3]{x^5}$ is not equivalent to the original expression $x^{-\frac{5}{3}}$. The negative sign in front of the expression does not change the fact that the base is raised to the power of $\frac{5}{3}$, which is not equivalent to the original expression.

Option D: $-\sqrt[5]{x^3}$

Option D: $-\sqrt[5]{x^3}$ is also not equivalent to the original expression $x^{-\frac{5}{3}}$. Similar to Option C, the negative sign in front of the expression does not change the fact that the base is raised to the power of $\frac{3}{5}$, which is not equivalent to the original expression.

Conclusion

In conclusion, the correct answer is Option B: $\frac{1}{\sqrt[3]{x^5}}$. This expression is equivalent to the original expression $x^{-\frac{5}{3}}$. Understanding how to simplify exponential expressions is crucial for solving various mathematical problems, and this article has provided a step-by-step guide on how to simplify the expression $x^{-\frac{5}{3}}$.

Additional Tips and Tricks

  • When simplifying exponential expressions, it's essential to understand the concept of negative exponents.
  • To simplify an expression with a negative exponent, rewrite it as the reciprocal of the base raised to the power of the exponent.
  • When comparing two expressions, look for equivalent forms by rewriting the expressions in a more familiar form.

Common Mistakes to Avoid

  • When simplifying exponential expressions, avoid making mistakes by not understanding the concept of negative exponents.
  • Avoid rewriting the expression in a way that changes the base or the exponent.
  • Avoid ignoring the negative sign in front of the expression.

Real-World Applications

Understanding how to simplify exponential expressions has numerous real-world applications. For example, in physics, exponential expressions are used to describe the behavior of particles and waves. In finance, exponential expressions are used to calculate compound interest. In computer science, exponential expressions are used to describe the time and space complexity of algorithms.

Final Thoughts

Q: What is the difference between a positive and negative exponent?

A: A positive exponent indicates that the base is raised to the power of the exponent, while a negative exponent indicates that the reciprocal of the base is raised to the power of the exponent.

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

A: To simplify an expression with a negative exponent, rewrite it as the reciprocal of the base raised to the power of the exponent. For example, $x^{-a} = \frac{1}{x^a}$.

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

A: When multiplying exponential expressions with the same base, add the exponents. For example, $x^a \cdot x^b = x^{a+b}$.

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

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

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

A: To simplify an expression with a fractional exponent, rewrite it as a product of two exponential expressions. For example, $x^{\frac{a}{b}} = (xa){\frac{1}{b}}$.

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

A: A radical expression is an expression that involves a root, such as $\sqrt{x}$, while an exponential expression is an expression that involves a power, such as $x^a$.

Q: How do I simplify an expression with a radical and an exponential?

A: To simplify an expression with a radical and an exponential, rewrite the radical as an exponential expression. For example, $\sqrt{x^a} = x^{\frac{a}{2}}$.

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

A: When simplifying an expression with a negative base, rewrite the expression as the reciprocal of the positive base raised to the power of the exponent. For example, $(-x)^a = -x^a$.

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

A: To simplify an expression with a variable base and a negative exponent, rewrite the expression as the reciprocal of the variable base raised to the power of the exponent. For example, $x^{-a} = \frac{1}{x^a}$.

Q: What is the rule for simplifying an expression with a variable base and a fractional exponent?

A: When simplifying an expression with a variable base and a fractional exponent, rewrite the expression as a product of two exponential expressions. For example, $x^{\frac{a}{b}} = (xa){\frac{1}{b}}$.

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

A: To simplify an expression with multiple bases and exponents, use the rules for multiplying and dividing exponential expressions with the same base, and then simplify the resulting expression.

Q: What is the rule for simplifying an expression with a zero exponent?

A: When simplifying an expression with a zero exponent, the result is always 1. For example, $x^0 = 1$.

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

A: To simplify an expression with a negative exponent and a variable base, rewrite the expression as the reciprocal of the variable base raised to the power of the exponent. For example, $x^{-a} = \frac{1}{x^a}$.

Q: What is the rule for simplifying an expression with a fractional exponent and a variable base?

A: When simplifying an expression with a fractional exponent and a variable base, rewrite the expression as a product of two exponential expressions. For example, $x^{\frac{a}{b}} = (xa){\frac{1}{b}}$.

Conclusion

Simplifying exponential expressions is a crucial skill for anyone who wants to excel in mathematics. By understanding the rules for simplifying exponential expressions, you can simplify complex expressions and solve various mathematical problems. Remember to always look for equivalent forms and avoid making common mistakes. With practice and patience, you will become proficient in simplifying exponential expressions and tackle even the most challenging mathematical problems.