Select The Correct Answer.Which Expression Is Equivalent To 5 8 \sqrt[8]{5} 8 5 ​ ?A. 5 3 \frac{5}{3} 3 5 ​ B. 5 3 5^3 5 3 C. 5 1 8 5^{\frac{1}{8}} 5 8 1 ​ D. 3 5 \frac{3}{5} 5 3 ​

Understanding Exponents and Roots: A Guide to Selecting the Correct Answer

When it comes to exponents and roots, it's essential to understand the relationship between them to solve problems accurately. In this article, we'll delve into the concept of roots, specifically the 8th root, and explore how it relates to exponents. We'll also examine the given options and determine which one is equivalent to $\sqrt[8]{5}$.

What is a Root?

A root is a mathematical operation that finds the value that, when raised to a certain power, gives a specified value. For example, the square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we're dealing with the 8th root, which means we're looking for a value that, when raised to the power of 8, gives the original number.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $2^3$ means $2 \times 2 \times 2$, which equals 8. Exponents can also be used to represent roots, where the exponent is the reciprocal of the root. For instance, $\sqrt[8]{5}$ can be written as $5^{\frac{1}{8}}$.

Evaluating the Options

Now that we've understood the concept of roots and exponents, let's evaluate the given options:

A. $\frac{5}{3}$

This option is not equivalent to $\sqrt[8]{5}$. The 8th root of 5 is a value that, when raised to the power of 8, gives 5. $\frac{5}{3}$ is a fraction that represents a value between 1 and 2, but it's not related to the 8th root of 5.

B. $5^3$

This option is also not equivalent to $\sqrt[8]{5}$. $5^3$ means $5 \times 5 \times 5$, which equals 125. This is not the same as the 8th root of 5.

C. $5^{\frac{1}{8}}$

This option is equivalent to $\sqrt[8]{5}$. As we discussed earlier, the 8th root of 5 can be written as $5^{\frac{1}{8}}$. This means that when $5^{\frac{1}{8}}$ is raised to the power of 8, it gives 5.

D. $\frac{3}{5}$

This option is not equivalent to $\sqrt[8]{5}$. $\frac{3}{5}$ is a fraction that represents a value between 0 and 1, but it's not related to the 8th root of 5.

In conclusion, the correct answer is C. $5^{\frac{1}{8}}$. This option is equivalent to $\sqrt[8]{5}$, as it represents the value that, when raised to the power of 8, gives 5. Understanding the relationship between exponents and roots is crucial in solving problems accurately, and this article has provided a comprehensive guide to help you select the correct answer.

Q: What is the difference between a root and an exponent?

A: A root is a mathematical operation that finds the value that, when raised to a certain power, gives a specified value. An exponent is a shorthand way of representing repeated multiplication.

Q: How do I evaluate the 8th root of a number?

A: To evaluate the 8th root of a number, you can use the formula $x^{\frac{1}{8}}$, where x is the number.

Q: What is the relationship between exponents and roots?

A: Exponents and roots are related in that the exponent is the reciprocal of the root. For example, $\sqrt[8]{5}$ can be written as $5^{\frac{1}{8}}$.

For further learning, we recommend the following resources:

• Khan Academy: Exponents and Roots
• Mathway: Exponents and Roots Calculator
• Wolfram Alpha: Exponents and Roots Calculator

By understanding the concept of roots and exponents, you'll be able to solve problems accurately and confidently. Remember to always evaluate the options carefully and use the correct formula to find the solution.
Exponents and Roots Q&A: A Comprehensive Guide

Exponents and roots are fundamental concepts in mathematics that are used to represent repeated multiplication and division. Understanding these concepts is crucial in solving problems accurately and confidently. In this article, we'll provide a comprehensive guide to exponents and roots, including frequently asked questions and additional resources.

Q: What is the difference between a root and an exponent?

A: A root is a mathematical operation that finds the value that, when raised to a certain power, gives a specified value. An exponent is a shorthand way of representing repeated multiplication.

Q: How do I evaluate the 8th root of a number?

A: To evaluate the 8th root of a number, you can use the formula $x^{\frac{1}{8}}$, where x is the number.

Q: What is the relationship between exponents and roots?

A: Exponents and roots are related in that the exponent is the reciprocal of the root. For example, $\sqrt[8]{5}$ can be written as $5^{\frac{1}{8}}$.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the following rules:

• When multiplying two numbers with the same base, add the exponents.
• When dividing two numbers with the same base, subtract the exponents.
• When raising a number to a power and then raising it to another power, multiply the exponents.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent represents a value that is raised to a power, while a negative exponent represents a value that is divided by a power.

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

A: To evaluate an expression with a negative exponent, you can use the following rule:

• $a^{-n} = \frac{1}{a^n}$

Q: What is the difference between a rational exponent and an irrational exponent?

A: A rational exponent is a fraction that represents a power, while an irrational exponent is a value that cannot be expressed as a fraction.

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

A: To evaluate an expression with a rational exponent, you can use the following rule:

• $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Q: What is the difference between a root and a power?

A: A root is a mathematical operation that finds the value that, when raised to a certain power, gives a specified value, while a power is a value that is raised to a certain power.

Q: How do I evaluate an expression with a root?

A: To evaluate an expression with a root, you can use the following rule:

• $\sqrt[n]{a} = a^{\frac{1}{n}}$

For further learning, we recommend the following resources:

• Khan Academy: Exponents and Roots
• Mathway: Exponents and Roots Calculator
• Wolfram Alpha: Exponents and Roots Calculator

To practice your skills, try solving the following problems:

• Evaluate the expression $2^3 \times 2^4$
• Simplify the expression $\sqrt[3]{8}$
• Evaluate the expression $5^{\frac{1}{2}}$
• Simplify the expression $\sqrt[4]{16}$

In conclusion, exponents and roots are fundamental concepts in mathematics that are used to represent repeated multiplication and division. Understanding these concepts is crucial in solving problems accurately and confidently. By following the rules and guidelines outlined in this article, you'll be able to evaluate expressions with exponents and roots with ease.

Q: What is the difference between a root and an exponent?

A: A root is a mathematical operation that finds the value that, when raised to a certain power, gives a specified value. An exponent is a shorthand way of representing repeated multiplication.

Q: How do I evaluate the 8th root of a number?

A: To evaluate the 8th root of a number, you can use the formula $x^{\frac{1}{8}}$, where x is the number.

Q: What is the relationship between exponents and roots?

A: Exponents and roots are related in that the exponent is the reciprocal of the root. For example, $\sqrt[8]{5}$ can be written as $5^{\frac{1}{8}}$.

• Forgetting to simplify expressions with exponents
• Not using the correct formula for evaluating roots
• Not following the rules for multiplying and dividing numbers with exponents

• Use the correct formula for evaluating roots
• Simplify expressions with exponents before evaluating them
• Use the rules for multiplying and dividing numbers with exponents to simplify expressions

In conclusion, exponents and roots are fundamental concepts in mathematics that are used to represent repeated multiplication and division. Understanding these concepts is crucial in solving problems accurately and confidently. By following the rules and guidelines outlined in this article, you'll be able to evaluate expressions with exponents and roots with ease.