Evaluate: 81 1 4 = 216 1 3 = \begin{array}{l} 81^{\frac{1}{4}}= \\ 216^{\frac{1}{3}}= \end{array} 8 1 4 1 ​ = 21 6 3 1 ​ = ​

Introduction

Exponents are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will delve into the world of exponents and explore how to evaluate expressions involving fractional exponents. We will examine two specific problems: $81^{\frac{1}{4}}$ and $216^{\frac{1}{3}}$, and provide step-by-step solutions to help you grasp the concept.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ can be read as "2 to the power of 3" or "2 cubed." It is equivalent to multiplying 2 by itself 3 times: $2^3 = 2 \times 2 \times 2 = 8$. Exponents can be positive, negative, or fractional, and they can be used to represent a wide range of mathematical operations.

Evaluating Exponents with Fractional Exponents

Fractional exponents are a type of exponent that involves a fraction as the exponent. They can be written in the form $a^{\frac{m}{n}}$, where $a$ is the base, $m$ is the numerator, and $n$ is the denominator. To evaluate an expression with a fractional exponent, we need to follow a specific set of rules.

Rule 1: Simplify the Fractional Exponent

The first step in evaluating an expression with a fractional exponent is to simplify the fraction. This involves dividing the numerator by the denominator to obtain a simplified fraction. For example, $\frac{1}{4}$ can be simplified to $\frac{1}{4}$, while $\frac{2}{3}$ can be simplified to $\frac{2}{3}$.

Rule 2: Apply the Power Rule

The power rule states that for any positive integer $n$, $(a^m)^n = a^{mn}$. This rule can be extended to fractional exponents by using the simplified fraction obtained in step 1. For example, $(a^{\frac{1}{4}})^4 = a^{\frac{1}{4} \times 4} = a^1 = a$.

Rule 3: Apply the Root Rule

The root rule states that for any positive integer $n$, $\sqrt[n]{a} = a^{\frac{1}{n}}$. This rule can be used to evaluate expressions involving fractional exponents by rewriting the expression in terms of a root. For example, $\sqrt[4]{81} = 81^{\frac{1}{4}}$.

Evaluating $81^{\frac{1}{4}}$

To evaluate $81^{\frac{1}{4}}$, we can use the root rule. We know that $\sqrt[4]{81} = 81^{\frac{1}{4}}$, so we can rewrite the expression as $\sqrt[4]{81}$. To evaluate this expression, we need to find the fourth root of 81.

Step 1: Find the Prime Factorization of 81

The prime factorization of 81 is $3^4$. This means that 81 can be expressed as the product of four 3's.

Step 2: Apply the Root Rule

Using the root rule, we can rewrite the expression as $\sqrt[4]{3^4}$. This means that we need to find the fourth root of $3^4$.

Step 3: Evaluate the Expression

To evaluate the expression, we can use the property of exponents that states $(a^m)^n = a^{mn}$. In this case, we have $(3^4)^{\frac{1}{4}} = 3^{4 \times \frac{1}{4}} = 3^1 = 3$.

Evaluating $216^{\frac{1}{3}}$

To evaluate $216^{\frac{1}{3}}$, we can use the root rule. We know that $\sqrt[3]{216} = 216^{\frac{1}{3}}$, so we can rewrite the expression as $\sqrt[3]{216}$. To evaluate this expression, we need to find the cube root of 216.

Step 1: Find the Prime Factorization of 216

The prime factorization of 216 is $2^3 \times 3^3$. This means that 216 can be expressed as the product of three 2's and three 3's.

Step 2: Apply the Root Rule

Using the root rule, we can rewrite the expression as $\sqrt[3]{2^3 \times 3^3}$. This means that we need to find the cube root of $2^3 \times 3^3$.

Step 3: Evaluate the Expression

To evaluate the expression, we can use the property of exponents that states $(a^m)^n = a^{mn}$. In this case, we have $\sqrt[3]{2^3 \times 3^3} = (2^3 \times 3^3)^{\frac{1}{3}} = 2^1 \times 3^1 = 2 \times 3 = 6$.

Conclusion

In this article, we have explored the concept of exponents and how to evaluate expressions involving fractional exponents. We have examined two specific problems: $81^{\frac{1}{4}}$ and $216^{\frac{1}{3}}$, and provided step-by-step solutions to help you grasp the concept. By following the rules outlined in this article, you should be able to evaluate expressions involving fractional exponents with confidence.

Final Thoughts

Q&A: Evaluating Exponents

Q: What is the difference between a positive exponent and a negative exponent? A: A positive exponent represents repeated multiplication, while a negative exponent represents repeated division. For example, $2^3 = 2 \times 2 \times 2 = 8$, while $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: How do I evaluate an expression with a fractional exponent? A: To evaluate an expression with a fractional exponent, you need to follow the rules outlined in the article. First, simplify the fraction, then apply the power rule, and finally apply the root rule.

Q: What is the power rule? A: The power rule states that for any positive integer $n$, $(a^m)^n = a^{mn}$. This rule can be extended to fractional exponents by using the simplified fraction obtained in step 1.

Q: What is the root rule? A: The root rule states that for any positive integer $n$, $\sqrt[n]{a} = a^{\frac{1}{n}}$. This rule can be used to evaluate expressions involving fractional exponents by rewriting the expression in terms of a root.

Q: How do I find the prime factorization of a number? A: To find the prime factorization of a number, you need to express the number as the product of its prime factors. For example, the prime factorization of 12 is $2^2 \times 3$.

Q: How do I apply the root rule to evaluate an expression? A: To apply the root rule, you need to rewrite the expression in terms of a root. For example, $\sqrt[3]{216} = 216^{\frac{1}{3}}$.

Q: What is the difference between a cube root and a fourth root? A: A cube root is the inverse operation of cubing, while a fourth root is the inverse operation of raising to the power of 4. For example, $\sqrt[3]{27} = 27^{\frac{1}{3}} = 3$, while $\sqrt[4]{81} = 81^{\frac{1}{4}} = 3$.

Q: How do I evaluate an expression with a negative exponent? A: To evaluate an expression with a negative exponent, you need to rewrite the expression in terms of a positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: What is the relationship between exponents and logarithms? A: Exponents and logarithms are inverse operations. For example, $2^3 = 8$ and $\log_2 8 = 3$.

Q: How do I use exponents to solve real-world problems? A: Exponents can be used to solve a wide range of real-world problems, such as calculating interest rates, modeling population growth, and analyzing data. For example, if you want to calculate the future value of an investment, you can use the formula $A = P(1 + r)^n$, where $A$ is the future value, $P$ is the principal amount, $r$ is the interest rate, and $n$ is the number of years.

Conclusion

In this article, we have provided a comprehensive guide to evaluating exponents, including a Q&A section to help you understand the concepts better. By mastering the concept of exponents, you will be able to tackle a wide range of mathematical challenges and develop a deeper understanding of mathematical concepts. So, take the time to practice evaluating exponents, and you will be well on your way to becoming a math whiz!

Final Thoughts

Exponents are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. By mastering the concept of exponents, you will be able to tackle a wide range of mathematical challenges and develop a deeper understanding of mathematical concepts. So, take the time to practice evaluating exponents, and you will be well on your way to becoming a math whiz!