Simplify.$81^{\frac{3}{4}}$
Introduction
In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with exponents, it's essential to understand the rules and properties that govern them. In this article, we will focus on simplifying the expression $81^{\frac{3}{4}}$. We will break down the problem step by step, using the properties of exponents and radicals to arrive at the final answer.
Understanding Exponents and Radicals
Before we dive into simplifying the expression, let's review the basics of exponents and radicals. An exponent is a small number that is raised to a power, indicating how many times a base number is multiplied by itself. For example, $2^3$ means 2 multiplied by itself three times, which equals 8. A radical, on the other hand, is a symbol that represents a number that can be expressed as a product of a whole number and a fraction. The most common radical is the square root, denoted by the symbol $\sqrt{}$.
Simplifying the Expression
Now that we have a basic understanding of exponents and radicals, let's focus on simplifying the expression $81^{\frac{3}{4}}$. To simplify this expression, we need to use the property of exponents that states $a^m \cdot a^n = a^{m+n}$. We can rewrite the expression as $\left(81{\frac{1}{4}}\right)3$.
Finding the Fourth Root of 81
To simplify the expression further, we need to find the fourth root of 81. The fourth root of a number is a value that, when raised to the power of 4, equals the original number. In this case, we need to find a number that, when raised to the power of 4, equals 81. We can start by listing the perfect fourth powers of numbers:
As we can see, the fourth power of 3 equals 81. Therefore, the fourth root of 81 is 3.
Simplifying the Expression Further
Now that we have found the fourth root of 81, we can simplify the expression further. We can rewrite the expression as $\left(3\right)^3$.
Evaluating the Expression
To evaluate the expression, we need to multiply 3 by itself three times. This equals 27.
Conclusion
In this article, we simplified the expression $81^{\frac{3}{4}}$ using the properties of exponents and radicals. We broke down the problem step by step, finding the fourth root of 81 and simplifying the expression further. We arrived at the final answer, which is 27. This problem demonstrates the importance of understanding the rules and properties of exponents and radicals in mathematics.
Common Mistakes to Avoid
When simplifying expressions involving exponents and radicals, there are several common mistakes to avoid. Here are a few:
- Not using the correct property of exponents: When simplifying expressions involving exponents, it's essential to use the correct property of exponents. In this case, we used the property $a^m \cdot a^n = a^{m+n}$.
- Not finding the correct root: When simplifying expressions involving radicals, it's essential to find the correct root. In this case, we found the fourth root of 81, which is 3.
- Not simplifying the expression further: When simplifying expressions, it's essential to simplify the expression further. In this case, we simplified the expression further by rewriting it as $\left(3\right)^3$.
Real-World Applications
Simplifying expressions involving exponents and radicals has several real-world applications. Here are a few:
- Science and Engineering: In science and engineering, simplifying expressions involving exponents and radicals is crucial for solving problems involving physical quantities such as distance, time, and velocity.
- Finance: In finance, simplifying expressions involving exponents and radicals is crucial for calculating interest rates, investment returns, and other financial metrics.
- Computer Science: In computer science, simplifying expressions involving exponents and radicals is crucial for solving problems involving algorithms, data structures, and software design.
Final Thoughts
Introduction
In our previous article, we simplified the expression $81^{\frac{3}{4}}$ using the properties of exponents and radicals. In this article, we will answer some frequently asked questions related to this problem.
Q: What is the difference between an exponent and a radical?
A: An exponent is a small number that is raised to a power, indicating how many times a base number is multiplied by itself. A radical, on the other hand, is a symbol that represents a number that can be expressed as a product of a whole number and a fraction.
Q: How do I simplify an expression involving exponents and radicals?
A: To simplify an expression involving exponents and radicals, you need to use the properties of exponents and radicals. In this case, we used the property $a^m \cdot a^n = a^{m+n}$ to simplify the expression.
Q: What is the fourth root of 81?
A: The fourth root of 81 is 3. This means that when 3 is raised to the power of 4, it equals 81.
Q: How do I find the fourth root of a number?
A: To find the fourth root of a number, you need to find a number that, when raised to the power of 4, equals the original number. In this case, we found that 3 raised to the power of 4 equals 81.
Q: What is the difference between $81^{\frac{3}{4}}$ and $\left(81{\frac{1}{4}}\right)3$?
A: $81^{\frac{3}{4}}$ and $\left(81{\frac{1}{4}}\right)3$ are equivalent expressions. The first expression can be rewritten as the second expression using the property $a^m \cdot a^n = a^{m+n}$.
Q: How do I evaluate an expression involving exponents and radicals?
A: To evaluate an expression involving exponents and radicals, you need to follow the order of operations (PEMDAS). In this case, we evaluated the expression $\left(3\right)^3$, which equals 27.
Q: What are some common mistakes to avoid when simplifying expressions involving exponents and radicals?
A: Some common mistakes to avoid when simplifying expressions involving exponents and radicals include:
- Not using the correct property of exponents
- Not finding the correct root
- Not simplifying the expression further
Q: What are some real-world applications of simplifying expressions involving exponents and radicals?
A: Some real-world applications of simplifying expressions involving exponents and radicals include:
- Science and engineering
- Finance
- Computer science
Conclusion
In this article, we answered some frequently asked questions related to simplifying the expression $81^{\frac{3}{4}}$. We covered topics such as the difference between an exponent and a radical, how to simplify an expression involving exponents and radicals, and some common mistakes to avoid. We also discussed some real-world applications of simplifying expressions involving exponents and radicals.
Additional Resources
For more information on simplifying expressions involving exponents and radicals, check out the following resources:
- Khan Academy: Exponents and Radicals
- Mathway: Exponents and Radicals
- Wolfram Alpha: Exponents and Radicals
Final Thoughts
In conclusion, simplifying expressions involving exponents and radicals is a crucial skill that helps us solve problems efficiently. By understanding the rules and properties of exponents and radicals, we can simplify complex expressions and arrive at the final answer.