Which Is Equivalent To $80^{\frac{1}{4} \times}$?A. $\left(\frac{80}{4}\right)^x$B. $\sqrt[4]{80}^x$C. $\sqrt[x]{80^4}$D. $\left(\frac{80}{x}\right)^4$

Introduction

Exponents and roots are fundamental concepts in mathematics that help us simplify complex expressions and solve equations. In this article, we will delve into the world of exponents and roots, exploring their properties and how they can be used to solve problems. We will also examine a specific problem that requires us to apply our knowledge of exponents and roots to find an equivalent expression.

What are Exponents and Roots?

Exponents and roots are two sides of the same coin. Exponents are used to represent repeated multiplication, while roots are used to represent repeated division. For example, the expression $2^3$ represents $2$ multiplied by itself $3$ times, while the expression $\sqrt[3]{8}$ represents the cube root of $8$, which is the number that, when multiplied by itself $3$ times, gives $8$.

Properties of Exponents

Exponents have several properties that make them useful in mathematics. Some of these properties include:

• Product of Powers: When we multiply two numbers with the same base, we can add their exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
• Power of a Power: When we raise a power to another power, we can multiply the exponents. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.
• Zero Exponent: Any number raised to the power of $0$ is equal to $1$. For example, $2^0 = 1$.

Properties of Roots

Roots also have several properties that make them useful in mathematics. Some of these properties include:

• Product of Roots: When we multiply two numbers with the same root, we can add their exponents. For example, $\sqrt[3]{2} \times \sqrt[3]{4} = \sqrt[3]{2 \times 4} = \sqrt[3]{8}$.
• Power of a Root: When we raise a root to another power, we can multiply the exponents. For example, $(\sqrt[3]{2})^4 = \sqrt[3]{2^4} = \sqrt[3]{16}$.
• Negative Exponent: Any number raised to the power of a negative exponent is equal to the reciprocal of the number raised to the positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Solving the Problem

Now that we have a good understanding of exponents and roots, let's apply our knowledge to solve the problem. The problem asks us to find an equivalent expression to $80^{\frac{1}{4} \times}$. To solve this problem, we need to use the properties of exponents and roots.

Step 1: Identify the Base and Exponent

The base of the expression is $80$, and the exponent is $\frac{1}{4}$. We can rewrite the expression as $80^{\frac{1}{4}}$.

Step 2: Apply the Property of Exponents

We can use the property of exponents to rewrite the expression as $\sqrt[4]{80}$.

Step 3: Simplify the Expression

The expression $\sqrt[4]{80}$ can be simplified by finding the fourth root of $80$. The fourth root of $80$ is a number that, when multiplied by itself $4$ times, gives $80$.

Conclusion

In conclusion, we have used the properties of exponents and roots to solve the problem and find an equivalent expression to $80^{\frac{1}{4} \times}$. The correct answer is $\sqrt[4]{80}^x$. We have also learned about the properties of exponents and roots, including the product of powers, power of a power, zero exponent, product of roots, power of a root, and negative exponent.

Final Answer

The final answer is $\boxed{\sqrt[4]{80}^x}$.

References

• Exponents and Roots
• Properties of Exponents
• Properties of Roots

Additional Resources

• Exponents and Roots Practice Problems
• Exponents and Roots Games

Frequently Asked Questions

• What is the difference between an exponent and a root?
  • An exponent represents repeated multiplication, while a root represents repeated division.
• How do I simplify an expression with a root?
  • To simplify an expression with a root, you can find the root of the number and multiply it by itself the required number of times.
• What is the property of exponents that states when we multiply two numbers with the same base, we can add their exponents?
  • The property of exponents that states when we multiply two numbers with the same base, we can add their exponents is called the product of powers.
    Exponents and Roots Q&A ==========================

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

A: An exponent represents repeated multiplication, while a root represents repeated division. For example, the expression $2^3$ represents $2$ multiplied by itself $3$ times, while the expression $\sqrt[3]{8}$ represents the cube root of $8$, which is the number that, when multiplied by itself $3$ times, gives $8$.

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

A: To simplify an expression with a root, you can find the root of the number and multiply it by itself the required number of times. For example, the expression $\sqrt[3]{8}$ can be simplified by finding the cube root of $8$, which is $2$. Therefore, $\sqrt[3]{8} = 2$.

Q: What is the property of exponents that states when we multiply two numbers with the same base, we can add their exponents?

A: The property of exponents that states when we multiply two numbers with the same base, we can add their exponents is called the product of powers. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.

Q: What is the property of exponents that states when we raise a power to another power, we can multiply the exponents?

A: The property of exponents that states when we raise a power to another power, we can multiply the exponents is called the power of a power. For example, $(2^3)^4 = 2^{3 \times 4} = 2^{12}$.

Q: What is the property of roots that states when we multiply two numbers with the same root, we can add their exponents?

A: The property of roots that states when we multiply two numbers with the same root, we can add their exponents is called the product of roots. For example, $\sqrt[3]{2} \times \sqrt[3]{4} = \sqrt[3]{2 \times 4} = \sqrt[3]{8}$.

Q: What is the property of roots that states when we raise a root to another power, we can multiply the exponents?

A: The property of roots that states when we raise a root to another power, we can multiply the exponents is called the power of a root. For example, $(\sqrt[3]{2})^4 = \sqrt[3]{2^4} = \sqrt[3]{16}$.

Q: What is the property of exponents that states any number raised to the power of $0$ is equal to $1$?

A: The property of exponents that states any number raised to the power of $0$ is equal to $1$ is called the zero exponent. For example, $2^0 = 1$.

Q: What is the property of roots that states any number raised to the power of a negative exponent is equal to the reciprocal of the number raised to the positive exponent?

A: The property of roots that states any number raised to the power of a negative exponent is equal to the reciprocal of the number raised to the positive exponent is called the negative exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

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

A: To simplify an expression with a negative exponent, you can rewrite the expression as the reciprocal of the number raised to the positive exponent. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Q: What is the final answer to the problem $80^{\frac{1}{4} \times}$?

A: The final answer to the problem $80^{\frac{1}{4} \times}$ is $\boxed{\sqrt[4]{80}^x}$.

Q: What are some additional resources for learning about exponents and roots?

A: Some additional resources for learning about exponents and roots include:

• Exponents and Roots Practice Problems
• Exponents and Roots Games
• Exponents and Roots Videos

Q: What are some frequently asked questions about exponents and roots?

A: Some frequently asked questions about exponents and roots include:

• What is the difference between an exponent and a root?
• How do I simplify an expression with a root?
• What is the property of exponents that states when we multiply two numbers with the same base, we can add their exponents?
• What is the property of exponents that states when we raise a power to another power, we can multiply the exponents?
• What is the property of roots that states when we multiply two numbers with the same root, we can add their exponents?
• What is the property of roots that states when we raise a root to another power, we can multiply the exponents?
• What is the property of exponents that states any number raised to the power of $0$ is equal to $1$?
• What is the property of roots that states any number raised to the power of a negative exponent is equal to the reciprocal of the number raised to the positive exponent?
• How do I simplify an expression with a negative exponent?
• What is the final answer to the problem $80^{\frac{1}{4} \times}$?