Evaluate:(-4/11)³

Mar 12, 2025 by ADMIN 19 views

15. Evaluate: (-4/11)³

Introduction

In this problem, we are required to evaluate the expression (-4/11)³, which involves cubing a fraction. To solve this, we need to understand the rules of exponentiation and how to handle fractions when raising them to a power.

Understanding Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. In this case, we are dealing with a fraction, (-4/11), and we need to raise it to the power of 3. To do this, we need to follow the rules of exponentiation, which state that:

When raising a fraction to a power, we raise the numerator and the denominator to that power separately.
When raising a negative number to an odd power, the result is negative.

Evaluating the Expression

To evaluate the expression (-4/11)³, we need to follow the rules of exponentiation. We will raise the numerator (-4) and the denominator (11) to the power of 3 separately.

Raising the Numerator to the Power of 3

To raise the numerator (-4) to the power of 3, we need to multiply it by itself three times:

(-4)³ = (-4) × (-4) × (-4) = -64

Raising the Denominator to the Power of 3

To raise the denominator (11) to the power of 3, we need to multiply it by itself three times:

(11)³ = 11 × 11 × 11 = 1331

Putting it All Together

Now that we have raised the numerator and the denominator to the power of 3, we can put them together to get the final result:

(-4/11)³ = (-64/1331)

Simplifying the Result

The result we obtained, (-64/1331), can be simplified further by dividing both the numerator and the denominator by their greatest common divisor (GCD). However, in this case, the GCD of 64 and 1331 is 1, so the result cannot be simplified further.

Conclusion

In conclusion, the value of (-4/11)³ is (-64/1331). This involves understanding the rules of exponentiation and how to handle fractions when raising them to a power.

Key Takeaways

When raising a fraction to a power, we raise the numerator and the denominator to that power separately.
When raising a negative number to an odd power, the result is negative.
The result of (-4/11)³ is (-64/1331).

Practice Problems

Evaluate the expression (2/3)⁴.
Evaluate the expression (-2/5)².
Evaluate the expression (3/4)⁵.

Solutions to Practice Problems

(2/3)⁴ = (16/81)
(-2/5)² = (4/25)
(3/4)⁵ = (243/1024)

Final Thoughts

In this problem, we learned how to evaluate the expression (-4/11)³ by following the rules of exponentiation. We also learned how to handle fractions when raising them to a power. These skills are essential in mathematics and can be applied to a wide range of problems.

Introduction

In the previous article, we evaluated the expression (-4/11)³ and obtained the result (-64/1331). In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on the topic.

Q&A

Q: What is the rule for raising a fraction to a power?

A: When raising a fraction to a power, we raise the numerator and the denominator to that power separately.

Q: What happens when we raise a negative number to an odd power?

A: When we raise a negative number to an odd power, the result is negative.

Q: Can we simplify the result (-64/1331)?

A: The result (-64/1331) cannot be simplified further because the greatest common divisor (GCD) of 64 and 1331 is 1.

Q: How do we evaluate the expression (2/3)⁴?

A: To evaluate the expression (2/3)⁴, we raise the numerator (2) and the denominator (3) to the power of 4 separately. This gives us (16/81).

Q: How do we evaluate the expression (-2/5)²?

A: To evaluate the expression (-2/5)², we raise the numerator (-2) and the denominator (5) to the power of 2 separately. This gives us (4/25).

Q: How do we evaluate the expression (3/4)⁵?

A: To evaluate the expression (3/4)⁵, we raise the numerator (3) and the denominator (4) to the power of 5 separately. This gives us (243/1024).

Q: What is the difference between an even and an odd power?

A: An even power is a power that is a multiple of 2, while an odd power is a power that is not a multiple of 2.

Q: How do we handle fractions when raising them to a power?

A: When raising a fraction to a power, we raise the numerator and the denominator to that power separately.

Q: Can we use a calculator to evaluate expressions with exponents?

A: Yes, we can use a calculator to evaluate expressions with exponents. However, it's always a good idea to understand the underlying math and be able to evaluate expressions by hand.

Conclusion

In this Q&A article, we provided answers to common questions related to evaluating expressions with exponents. We also provided additional information on the topic and clarified any doubts. By understanding the rules of exponentiation and how to handle fractions when raising them to a power, we can evaluate expressions with exponents with confidence.

Key Takeaways

When raising a fraction to a power, we raise the numerator and the denominator to that power separately.
When raising a negative number to an odd power, the result is negative.
The result of (-4/11)³ is (-64/1331).
We can use a calculator to evaluate expressions with exponents, but it's always a good idea to understand the underlying math.