Simplify The Expression:$(-64)^{-\frac{2}{3}}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common types of expressions that require simplification is those involving exponents and roots. In this article, we will focus on simplifying the expression $(-64)^{-\frac{2}{3}}$. We will break down the problem step by step, using the properties of exponents and roots to simplify the expression.

Understanding the Expression

The given expression is $(-64)^{-\frac{2}{3}}$. To simplify this expression, we need to understand the properties of exponents and roots. The exponent $-\frac{2}{3}$ indicates that we need to take the cube root of $-64$ and then raise the result to the power of $-2$.

Simplifying the Expression

To simplify the expression, we can start by taking the cube root of $-64$. The cube root of $-64$ is $-4$, since $(-4)^3 = -64$. Now, we need to raise $-4$ to the power of $-2$.

Using the Property of Negative Exponents

When we raise a number to a negative power, we can rewrite it as the reciprocal of the number raised to the positive power. In other words, $a^{-n} = \frac{1}{a^n}$. Using this property, we can rewrite the expression as:

$$(-64)^{-\frac{2}{3}} = \frac{1}{(-4)^2}$$

Simplifying the Expression Further

Now, we can simplify the expression further by evaluating the denominator. The denominator is $(-4)^2$, which is equal to $16$. Therefore, the expression becomes:

$$\frac{1}{16}$$

Conclusion

In this article, we simplified the expression $(-64)^{-\frac{2}{3}}$ using the properties of exponents and roots. We started by taking the cube root of $-64$ and then raised the result to the power of $-2$. Using the property of negative exponents, we rewrote the expression as the reciprocal of the number raised to the positive power. Finally, we simplified the expression further by evaluating the denominator.

Frequently Asked Questions

• What is the cube root of $-64$?
• How do we simplify an expression with a negative exponent?
• What is the property of negative exponents?

Step-by-Step Solution

1. Take the cube root of $-64$.
2. Raise the result to the power of $-2$.
3. Rewrite the expression using the property of negative exponents.
4. Simplify the expression further by evaluating the denominator.

Common Mistakes

• Forgetting to take the cube root of $-64$.
• Not using the property of negative exponents.
• Not simplifying the expression further by evaluating the denominator.

Real-World Applications

Simplifying expressions like $(-64)^{-\frac{2}{3}}$ is crucial in many real-world applications, such as:

• Physics: Simplifying expressions involving exponents and roots is essential in physics, where we often encounter complex mathematical problems.
• Engineering: Engineers use mathematical models to design and optimize systems, and simplifying expressions is a crucial step in this process.
• Computer Science: Simplifying expressions is a fundamental skill in computer science, where we often encounter complex mathematical problems in algorithms and data structures.

Final Thoughts

Simplifying expressions like $(-64)^{-\frac{2}{3}}$ requires a deep understanding of the properties of exponents and roots. By following the steps outlined in this article, we can simplify complex expressions and solve problems efficiently and accurately. Whether you are a student, a professional, or simply someone who enjoys mathematics, simplifying expressions is a valuable skill that can help you solve problems and achieve your goals.

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Introduction

In our previous article, we simplified the expression $(-64)^{-\frac{2}{3}}$ using the properties of exponents and roots. In this article, we will answer some frequently asked questions related to simplifying expressions like $(-64)^{-\frac{2}{3}}$. We will cover topics such as the cube root of $-64$, simplifying expressions with negative exponents, and real-world applications of simplifying expressions.

Q&A

Q: What is the cube root of $-64$?

A: The cube root of $-64$ is $-4$, since $(-4)^3 = -64$.

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

A: To simplify an expression with a negative exponent, we can rewrite it as the reciprocal of the number raised to the positive power. In other words, $a^{-n} = \frac{1}{a^n}$.

Q: What is the property of negative exponents?

A: The property of negative exponents states that $a^{-n} = \frac{1}{a^n}$. This means that we can rewrite an expression with a negative exponent as the reciprocal of the number raised to the positive power.

Q: How do we simplify the expression $(-64)^{-\frac{2}{3}}$?

A: To simplify the expression $(-64)^{-\frac{2}{3}}$, we can follow these steps:

1. Take the cube root of $-64$.
2. Raise the result to the power of $-2$.
3. Rewrite the expression using the property of negative exponents.
4. Simplify the expression further by evaluating the denominator.

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

A: Simplifying expressions is crucial in many real-world applications, such as:

• Physics: Simplifying expressions involving exponents and roots is essential in physics, where we often encounter complex mathematical problems.
• Engineering: Engineers use mathematical models to design and optimize systems, and simplifying expressions is a crucial step in this process.
• Computer Science: Simplifying expressions is a fundamental skill in computer science, where we often encounter complex mathematical problems in algorithms and data structures.

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

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

• Forgetting to take the cube root of $-64$.
• Not using the property of negative exponents.
• Not simplifying the expression further by evaluating the denominator.

Step-by-Step Solution

1. Take the cube root of $-64$.
2. Raise the result to the power of $-2$.
3. Rewrite the expression using the property of negative exponents.
4. Simplify the expression further by evaluating the denominator.

Real-World Applications

Simplifying expressions like $(-64)^{-\frac{2}{3}}$ is crucial in many real-world applications, such as:

• Physics: Simplifying expressions involving exponents and roots is essential in physics, where we often encounter complex mathematical problems.
• Engineering: Engineers use mathematical models to design and optimize systems, and simplifying expressions is a crucial step in this process.
• Computer Science: Simplifying expressions is a fundamental skill in computer science, where we often encounter complex mathematical problems in algorithms and data structures.

Final Thoughts

Simplifying expressions like $(-64)^{-\frac{2}{3}}$ requires a deep understanding of the properties of exponents and roots. By following the steps outlined in this article, we can simplify complex expressions and solve problems efficiently and accurately. Whether you are a student, a professional, or simply someone who enjoys mathematics, simplifying expressions is a valuable skill that can help you solve problems and achieve your goals.

Additional Resources

• Khan Academy: Exponents and Roots
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions like $(-64)^{-\frac{2}{3}}$. We covered topics such as the cube root of $-64$, simplifying expressions with negative exponents, and real-world applications of simplifying expressions. By following the steps outlined in this article, we can simplify complex expressions and solve problems efficiently and accurately.