Evaluate: $216^{\frac{1}{3}}$A) -3 B) 9 C) 6 D) 1

Introduction

In mathematics, exponents and roots are fundamental concepts that help us solve various types of problems. One of the most common operations involving exponents and roots is evaluating expressions with fractional exponents. In this article, we will focus on evaluating the expression $216^{\frac{1}{3}}$ and determine the correct answer among the given options.

Understanding Fractional Exponents

Before we dive into evaluating the expression, let's briefly review what fractional exponents represent. A fractional exponent is a way of expressing a root in terms of an exponent. For example, $a^{\frac{1}{n}}$ is equivalent to the nth root of a, denoted as $\sqrt[n]{a}$. In this case, the expression $216^{\frac{1}{3}}$ can be rewritten as the cube root of 216.

Evaluating the Expression

To evaluate the expression $216^{\frac{1}{3}}$, we need to find the cube root of 216. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In other words, if $x = \sqrt[3]{216}$, then $x^3 = 216$. We can find the cube root of 216 by factoring it into prime factors.

Factoring 216

To factor 216, we can start by breaking it down into its prime factors. We know that 216 is a multiple of 2, so we can start by dividing it by 2. This gives us:

$216 = 2 \times 108$

We can continue factoring 108 by dividing it by 2 again:

$108 = 2 \times 54$

$54 = 2 \times 27$

$27 = 3 \times 9$

$9 = 3 \times 3$

Now that we have factored 216 into its prime factors, we can rewrite it as:

$216 = 2^3 \times 3^3$

Finding the Cube Root

Now that we have factored 216 into its prime factors, we can find the cube root by taking the cube root of each prime factor. Since the cube root of $2^3$ is 2 and the cube root of $3^3$ is 3, we can conclude that the cube root of 216 is:

$\sqrt[3]{216} = 2 \times 3 = 6$

Conclusion

In conclusion, the expression $216^{\frac{1}{3}}$ can be evaluated by finding the cube root of 216. By factoring 216 into its prime factors and taking the cube root of each prime factor, we can conclude that the cube root of 216 is 6. Therefore, the correct answer among the given options is:

The final answer is C) 6

Introduction

In our previous article, we evaluated the expression $216^{\frac{1}{3}}$ and determined that the correct answer is C) 6. However, we understand that some readers may still have questions or doubts about the concept of fractional exponents and how to evaluate expressions with them. In this article, we will address some of the most frequently asked questions about evaluating expressions with fractional exponents.

Q&A

Q: What is a fractional exponent?

A: A fractional exponent is a way of expressing a root in terms of an exponent. For example, $a^{\frac{1}{n}}$ is equivalent to the nth root of a, denoted as $\sqrt[n]{a}$.

Q: How do I evaluate an expression with a fractional exponent?

A: To evaluate an expression with a fractional exponent, you need to find the root of the number. For example, to evaluate $216^{\frac{1}{3}}$, you need to find the cube root of 216.

Q: What is the difference between a fractional exponent and a decimal exponent?

A: A fractional exponent is a way of expressing a root in terms of an exponent, while a decimal exponent is a way of expressing a power in terms of an exponent. For example, $a^{\frac{1}{3}}$ is equivalent to the cube root of a, while $a^{0.5}$ is equivalent to the square root of a.

Q: Can I simplify an expression with a fractional exponent?

A: Yes, you can simplify an expression with a fractional exponent by finding the root of the number and simplifying the expression. For example, $216^{\frac{1}{3}}$ can be simplified to 6.

Q: What are some common fractional exponents?

A: Some common fractional exponents include:

• $a^{\frac{1}{2}}$ = square root of a
• $a^{\frac{1}{3}}$ = cube root of a
• $a^{\frac{1}{4}}$ = fourth root of a
• $a^{\frac{1}{5}}$ = fifth root of a

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

A: To evaluate an expression with a negative fractional exponent, you need to find the reciprocal of the number and then find the root of the reciprocal. For example, to evaluate $216^{-\frac{1}{3}}$, you need to find the reciprocal of 216 and then find the cube root of the reciprocal.

Q: Can I use a calculator to evaluate an expression with a fractional exponent?

A: Yes, you can use a calculator to evaluate an expression with a fractional exponent. However, make sure to use the correct function on the calculator, such as the "root" or "power" function.

Conclusion

In conclusion, evaluating expressions with fractional exponents can be a bit challenging, but with practice and patience, you can become proficient in evaluating them. Remember to always follow the order of operations and to simplify the expression as much as possible. If you have any more questions or doubts, feel free to ask.

The final answer is C) 6