Simplify The Expression: $ \left(27 X^9\right)^{-2 / 3} $

Understanding the Problem

The given expression is $ \left(27 x^9\right){-2 / 3} $. To simplify this expression, we need to apply the rules of exponents and understand the properties of negative exponents. Negative exponents are a way of expressing a fraction as an exponent. For example, $ a^{-n} = \frac{1}{a^n} $. We will use this property to simplify the given expression.

Step 1: Break Down the Expression

The expression $ \left(27 x^9\right){-2 / 3} $ can be broken down into two parts: $ 27 $ and $ x^9 $. We will simplify each part separately and then combine them.

Simplifying the Coefficient

The coefficient $ 27 $ can be written as $ 3^3 $. This is because $ 27 = 3 \times 3 \times 3 $. We can rewrite the expression as $ \left(3^3 x^9\right){-2 / 3} $.

Simplifying the Variable

The variable part $ x^9 $ can be rewritten as $ x^{9 \times (-2/3)} $ using the property of exponents $ (a^m)n = a^{m \times n} $. This simplifies to $ x^{-6} $.

Combining the Parts

Now that we have simplified the coefficient and the variable, we can combine them to get the final expression. $ \left(3^3 x^9\right){-2 / 3} = (3³⁾{-2/3} \times (x⁹⁾{-2/3} = 3^{-2} \times x^{-6} = \frac{1}{3^2} \times x^{-6} = \frac{1}{9} \times x^{-6} = \frac{x^{-6}}{9} = \frac{1}{9x^6} $.

Conclusion

In conclusion, the simplified expression is $ \frac{1}{9x^6} $. This is the final answer to the given problem.

Key Takeaways

• Negative exponents can be expressed as fractions.
• The property of exponents $ (a^m)n = a^{m \times n} $ can be used to simplify expressions.
• The expression $ \left(27 x^9\right){-2 / 3} $ can be simplified to $ \frac{1}{9x^6} $.

Real-World Applications

Simplifying expressions with negative exponents has many real-world applications. For example, in physics, the expression $ \frac{1}{9x^6} $ can be used to describe the motion of an object under the influence of gravity. In engineering, the expression can be used to design systems that involve negative feedback.

Common Mistakes

When simplifying expressions with negative exponents, it's easy to make mistakes. Some common mistakes include:

• Forgetting to apply the property of exponents $ (a^m)n = a^{m \times n} $.
• Not simplifying the coefficient and the variable separately.
• Not combining the parts correctly.

Tips and Tricks

To simplify expressions with negative exponents, follow these tips and tricks:

• Break down the expression into two parts: the coefficient and the variable.
• Simplify each part separately using the properties of exponents.
• Combine the parts correctly to get the final expression.

Practice Problems

To practice simplifying expressions with negative exponents, try the following problems:

• Simplify the expression $ \left(16 x^4\right){-3 / 2} $.
• Simplify the expression $ \left(25 x^6\right){-4 / 3} $.
• Simplify the expression $ \left(9 x^8\right){-5 / 4} $.

Conclusion

Q: What is a negative exponent?

A: A negative exponent is a way of expressing a fraction as an exponent. For example, $ a^{-n} = \frac{1}{a^n} $. Negative exponents can be simplified using the property of exponents $ (a^m)n = a^{m \times n} $.

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

A: To simplify an expression with a negative exponent, follow these steps:

1. Break down the expression into two parts: the coefficient and the variable.
2. Simplify each part separately using the properties of exponents.
3. Combine the parts correctly to get the final expression.

Q: What is the property of exponents $ (a^m)n = a^{m \times n} $?

A: The property of exponents $ (a^m)n = a^{m \times n} $ states that when a power is raised to a power, the exponents are multiplied. For example, $ (a^m)n = a^{m \times n} $.

Q: How do I simplify the expression $ \left(27 x^9\right){-2 / 3} $?

A: To simplify the expression $ \left(27 x^9\right){-2 / 3} $, follow these steps:

1. Break down the expression into two parts: the coefficient and the variable.
2. Simplify the coefficient: $ 27 = 3^3 $.
3. Simplify the variable: $ x^9 = x^{9 \times (-2/3)} = x^{-6} $.
4. Combine the parts correctly to get the final expression: $ \frac{1}{9x^6} $.

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

A: Some common mistakes to avoid when simplifying expressions with negative exponents include:

• Forgetting to apply the property of exponents $ (a^m)n = a^{m \times n} $.
• Not simplifying the coefficient and the variable separately.
• Not combining the parts correctly.

Q: How do I practice simplifying expressions with negative exponents?

A: To practice simplifying expressions with negative exponents, try the following problems:

• Simplify the expression $ \left(16 x^4\right){-3 / 2} $.
• Simplify the expression $ \left(25 x^6\right){-4 / 3} $.
• Simplify the expression $ \left(9 x^8\right){-5 / 4} $.

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

A: Simplifying expressions with negative exponents has many real-world applications, including:

• Physics: The expression $ \frac{1}{9x^6} $ can be used to describe the motion of an object under the influence of gravity.
• Engineering: The expression can be used to design systems that involve negative feedback.

Q: How do I use the property of exponents $ (a^m)n = a^{m \times n} $ to simplify expressions with negative exponents?

A: To use the property of exponents $ (a^m)n = a^{m \times n} $ to simplify expressions with negative exponents, follow these steps:

1. Break down the expression into two parts: the coefficient and the variable.
2. Simplify each part separately using the properties of exponents.
3. Combine the parts correctly to get the final expression.

Conclusion

In conclusion, simplifying expressions with negative exponents is an important skill in mathematics. By understanding the properties of negative exponents and following the tips and tricks outlined in this article, you can simplify expressions with ease. Practice problems are also provided to help you reinforce your understanding of the concept.