Which Is Equivalent To $80^{\frac{1}{4} X}$?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 are used to represent repeated multiplication and nth roots of numbers. In this article, we will delve into the world of exponents and roots, exploring their properties, rules, and applications. We will also examine a specific problem involving exponents and roots, and determine which of the given options is equivalent to the expression $80^{\frac{1}{4} x}$.

What are Exponents and Roots?

Exponents and roots are used to represent repeated multiplication and nth roots of numbers. An exponent is a small number that is raised to a power, indicating how many times the base number is multiplied by itself. For example, $2^3$ represents 2 multiplied by itself 3 times, or $2 \times 2 \times 2 = 8$. On the other hand, a root is a number that is raised to a power, indicating how many times the base number is divided by itself. For example, $\sqrt[3]{8}$ represents the cube root of 8, or the number that, when multiplied by itself 3 times, equals 8.

Properties of Exponents

Exponents have several properties that are essential to understand. These properties include:

• Product of Powers: When multiplying two numbers with the same base, we add their exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
• Power of a Power: When raising a number with an exponent to another power, we multiply the exponents. For example, $(2³⁾4 = 2^{3 \times 4} = 2^{12}$.
• Zero Exponent: Any number raised to the power of 0 is equal to 1. For example, $2^0 = 1$.
• Negative Exponent: A negative exponent represents the reciprocal of the number. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Properties of Roots

Roots also have several properties that are essential to understand. These properties include:

• Product of Roots: When multiplying two numbers with the same base, we multiply their roots. For example, $\sqrt{2} \times \sqrt{3} = \sqrt{2 \times 3} = \sqrt{6}$.
• Power of a Root: When raising a number with a root to another power, we multiply the roots. For example, $(\sqrt{2})^4 = 2^{4/2} = 2^2$.
• Zero Root: Any number raised to the power of 0 is equal to 1. For example, $\sqrt[3]{2^0} = 1$.
• Negative Root: A negative root represents the reciprocal of the number. For example, $\sqrt[3]{-2} = -\sqrt[3]{2}$.

Solving the Problem

Now that we have a solid understanding of exponents and roots, let's tackle the problem at hand. We are given the expression $80^{\frac{1}{4} x}$ and need to determine which of the given options is equivalent to it.

Option A: $\left(\frac{80}{4}\right)^x$

This option is incorrect because it does not take into account the exponent $\frac{1}{4} x$. The expression $\left(\frac{80}{4}\right)^x$ represents $20^x$, which is not equivalent to $80^{\frac{1}{4} x}$.

Option B: $\sqrt[4]{80}^x$

This option is also incorrect because it does not take into account the exponent $\frac{1}{4} x$. The expression $\sqrt[4]{80}^x$ represents $(2⁴⁾x = 2^{4x}$, which is not equivalent to $80^{\frac{1}{4} x}$.

Option C: $\sqrt[x]{80^4}$

This option is correct because it takes into account the exponent $\frac{1}{4} x$. The expression $\sqrt[x]{80^4}$ represents $\sqrt[x]{(2⁴⁾x} = \sqrt[x]{2^{4x}} = 2^{\frac{4x}{x}} = 2^4 = 80$, which is equivalent to $80^{\frac{1}{4} x}$.

Option D: $\left(\frac{80}{x}\right)^4$

This option is incorrect because it does not take into account the exponent $\frac{1}{4} x$. The expression $\left(\frac{80}{x}\right)^4$ represents $\left(\frac{2^4}{x}\right)4 = \frac{(2⁴⁾4}{x^4} = \frac{2^{16}}{x4}$, which is not equivalent to $80^{\frac{1}{4} x}$.

Conclusion

In conclusion, the correct answer to the problem is Option C: $\sqrt[x]{80^4}$. This option takes into account the exponent $\frac{1}{4} x$ and is equivalent to the expression $80^{\frac{1}{4} x}$. We hope that this article has provided a comprehensive guide to understanding exponents and roots, and has helped to clarify the properties and rules of these mathematical concepts.

Introduction

Exponents and roots are fundamental concepts in mathematics that are used to represent repeated multiplication and nth roots of numbers. In our previous article, we explored the properties and rules of exponents and roots, and solved a problem involving the expression $80^{\frac{1}{4} x}$. In this article, we will continue to delve into the world of exponents and roots, answering some of the most frequently asked questions about these mathematical concepts.

Q&A

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

A: An exponent is a small number that is raised to a power, indicating how many times the base number is multiplied by itself. A root, on the other hand, is a number that is raised to a power, indicating how many times the base number is divided by itself.

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

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

• Product of Powers: When multiplying two numbers with the same base, you add their exponents. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.
• Power of a Power: When raising a number with an exponent to another power, you multiply the exponents. For example, $(2³⁾4 = 2^{3 \times 4} = 2^{12}$.
• Zero Exponent: Any number raised to the power of 0 is equal to 1. For example, $2^0 = 1$.
• Negative Exponent: A negative exponent represents the reciprocal of the number. For example, $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

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

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

• Product of Roots: When multiplying two numbers with the same base, you multiply their roots. For example, $\sqrt{2} \times \sqrt{3} = \sqrt{2 \times 3} = \sqrt{6}$.
• Power of a Root: When raising a number with a root to another power, you multiply the roots. For example, $(\sqrt{2})^4 = 2^{4/2} = 2^2$.
• Zero Root: Any number raised to the power of 0 is equal to 1. For example, $\sqrt[3]{2^0} = 1$.
• Negative Root: A negative root represents the reciprocal of the number. For example, $\sqrt[3]{-2} = -\sqrt[3]{2}$.

Q: How do I evaluate an expression with a variable in the exponent?

A: To evaluate an expression with a variable in the exponent, you can use the following rules:

• Exponent Rule: When evaluating an expression with a variable in the exponent, you can use the exponent rule to simplify the expression. For example, $2^{3x} = (2³⁾x = 8^x$.
• Power Rule: When evaluating an expression with a variable in the exponent, you can use the power rule to simplify the expression. For example, $(2^x)3 = 2^{3x}$.

Q: How do I solve an equation with an exponent?

A: To solve an equation with an exponent, you can use the following steps:

• Isolate the Exponent: Isolate the exponent on one side of the equation.
• Use the Exponent Rule: Use the exponent rule to simplify the equation.
• Solve for the Variable: Solve for the variable by isolating it on one side of the equation.

Q: How do I solve an equation with a root?

A: To solve an equation with a root, you can use the following steps:

• Isolate the Root: Isolate the root on one side of the equation.
• Use the Root Rule: Use the root rule to simplify the equation.
• Solve for the Variable: Solve for the variable by isolating it on one side of the equation.

Conclusion

In conclusion, exponents and roots are fundamental concepts in mathematics that are used to represent repeated multiplication and nth roots of numbers. In this article, we have answered some of the most frequently asked questions about exponents and roots, and provided guidance on how to simplify expressions with exponents and roots, evaluate expressions with variables in the exponent, and solve equations with exponents and roots. We hope that this article has provided a comprehensive guide to understanding exponents and roots, and has helped to clarify the properties and rules of these mathematical concepts.