Find The Indicated Real { N $}$th Root(s) Of { A $} . . . { N=3, \ A=-216 \} A. -72 B. -6 C. { -\frac{1}{6}$}$ D. 6
Introduction
In mathematics, finding the nth root of a number is a fundamental operation that involves determining the value of a number that, when raised to a given power, equals the original number. In this article, we will focus on finding the indicated real nth root of a number, specifically the cube root of -216.
Understanding the Problem
The problem asks us to find the cube root of -216, which means we need to determine the value of x such that x^3 = -216. To solve this problem, we can use various mathematical techniques, including factoring, prime factorization, and the use of mathematical formulas.
Factoring -216
To begin solving the problem, we can factor -216 into its prime factors. The prime factorization of -216 is:
-216 = -2^3 * 3^3
Using the Formula for nth Roots
The formula for finding the nth root of a number is:
x = ∛a = a^(1/n)
where a is the number and n is the power to which we want to raise the number.
In this case, we want to find the cube root of -216, so we can use the formula:
x = ∛(-216) = (-216)^(1/3)
Simplifying the Expression
To simplify the expression, we can use the fact that (-216)^(1/3) = (-2^3 * 33)(1/3). Using the properties of exponents, we can rewrite this expression as:
(-2^3 * 33)(1/3) = (-2) * (3) = -6
Evaluating the Options
Now that we have found the cube root of -216, we can evaluate the options given in the problem:
A. -72 B. -6 C. -1/6 D. 6
Based on our calculation, we can see that option B, -6, is the correct answer.
Conclusion
In this article, we have discussed how to find the indicated real nth root of a number, specifically the cube root of -216. We used various mathematical techniques, including factoring and the use of mathematical formulas, to solve the problem. By following these steps, we were able to determine that the cube root of -216 is -6.
Additional Tips and Tricks
- When finding the nth root of a number, it is essential to use the correct formula and to simplify the expression carefully.
- Factoring and prime factorization can be useful tools in solving problems involving nth roots.
- The use of mathematical formulas and properties of exponents can help simplify complex expressions and make it easier to find the nth root of a number.
Common Mistakes to Avoid
- When finding the nth root of a number, it is easy to make mistakes by not using the correct formula or by not simplifying the expression carefully.
- Failing to factor or prime factorize the number can lead to incorrect solutions.
- Not using the properties of exponents can make it difficult to simplify complex expressions and find the nth root of a number.
Real-World Applications
Finding the nth root of a number has many real-world applications, including:
- Calculating interest rates and investments
- Determining the volume of a cube or rectangular prism
- Finding the area of a square or rectangle
- Solving problems involving exponential growth and decay
Final Thoughts
Introduction
In our previous article, we discussed how to find the indicated real nth root of a number, specifically the cube root of -216. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on this topic.
Q: What is the formula for finding the nth root of a number?
A: The formula for finding the nth root of a number is:
x = ∛a = a^(1/n)
where a is the number and n is the power to which we want to raise the number.
Q: How do I simplify the expression for the nth root of a number?
A: To simplify the expression for the nth root of a number, you can use the properties of exponents and the fact that (-a)^(1/n) = -a^(1/n). You can also use factoring and prime factorization to simplify the expression.
Q: What are some common mistakes to avoid when finding the nth root of a number?
A: Some common mistakes to avoid when finding the nth root of a number include:
- Not using the correct formula
- Not simplifying the expression carefully
- Failing to factor or prime factorize the number
- Not using the properties of exponents
Q: How do I determine the value of n in the formula for the nth root of a number?
A: To determine the value of n in the formula for the nth root of a number, you need to know the power to which you want to raise the number. For example, if you want to find the cube root of a number, n = 3.
Q: Can I use a calculator to find the nth root of a number?
A: Yes, you can use a calculator to find the nth root of a number. However, it is essential to understand the formula and the properties of exponents to ensure that you are using the calculator correctly.
Q: How do I apply the formula for the nth root of a number to real-world problems?
A: The formula for the nth root of a number can be applied to various real-world problems, including:
- Calculating interest rates and investments
- Determining the volume of a cube or rectangular prism
- Finding the area of a square or rectangle
- Solving problems involving exponential growth and decay
Q: What are some additional tips and tricks for finding the nth root of a number?
A: Some additional tips and tricks for finding the nth root of a number include:
- Using factoring and prime factorization to simplify the expression
- Applying the properties of exponents to simplify the expression
- Using a calculator to check your work
- Practicing problems involving nth roots to build your skills and confidence
Q: Can I find the nth root of a negative number?
A: Yes, you can find the nth root of a negative number. However, it is essential to understand that the nth root of a negative number may be a complex number.
Q: How do I determine if the nth root of a number is a real or complex number?
A: To determine if the nth root of a number is a real or complex number, you need to examine the expression for the nth root. If the expression involves a negative number raised to an odd power, the nth root may be a complex number.
Conclusion
In this Q&A article, we have provided additional information and clarification on the topic of finding the indicated real nth root of a number. We hope that this article has been helpful in answering your questions and providing you with a better understanding of this topic.