Which Expressions Are Equivalent To $\sqrt[3]{128}$? Select Three Correct Answers.A. $128^{\frac{\pi}{3}}$B. $128^{\frac{2}{x}}$C. $(4 \sqrt[3]{2})^x$D. $\left(4\left(2^{\frac{1}{3}}\right)\right)^x$E.

Introduction

Radical expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $\sqrt[3]{128}$ and identifying equivalent expressions. We will explore three correct answers and provide a detailed explanation of each.

Understanding the Expression

The given expression is $\sqrt[3]{128}$. To simplify this expression, we need to find the cube root of 128. The cube root of a number is a value that, when multiplied by itself three times, gives the original number.

Simplifying the Expression

To simplify the expression $\sqrt[3]{128}$, we can start by finding the prime factorization of 128.

$$128 = 2^7$$

Now, we can rewrite the expression as:

$$\sqrt[3]{128} = \sqrt[3]{2^7}$$

Using the property of radicals, we can rewrite the expression as:

$$\sqrt[3]{2^7} = 2^{\frac{7}{3}}$$

Equivalent Expressions

Now that we have simplified the expression $\sqrt[3]{128}$, we can identify equivalent expressions. We will focus on three correct answers: A, C, and D.

A. $128^{\frac{\pi}{3}}$

This expression is equivalent to the original expression $\sqrt[3]{128}$. To see why, we can rewrite the expression as:

$$128^{\frac{\pi}{3}} = (2^7)^{\frac{\pi}{3}}$$

Using the property of exponents, we can rewrite the expression as:

$$(2^7)^{\frac{\pi}{3}} = 2^{\frac{7\pi}{3}}$$

Since $\pi$ is an irrational number, we cannot simplify the expression further. However, we can see that the expression is equivalent to the original expression $\sqrt[3]{128}$.

C. $(4 \sqrt[3]{2})^x$

This expression is equivalent to the original expression $\sqrt[3]{128}$. To see why, we can rewrite the expression as:

$$(4 \sqrt[3]{2})^x = (4 \cdot 2^{\frac{1}{3}})^x$$

Using the property of exponents, we can rewrite the expression as:

$$(4 \cdot 2^{\frac{1}{3}})^x = 4^x \cdot 2^{\frac{x}{3}}$$

Since $4 = 2^2$, we can rewrite the expression as:

$$4^x \cdot 2^{\frac{x}{3}} = (2^2)^x \cdot 2^{\frac{x}{3}}$$

Using the property of exponents, we can rewrite the expression as:

$$(2^2)^x \cdot 2^{\frac{x}{3}} = 2^{2x} \cdot 2^{\frac{x}{3}}$$

Using the property of exponents, we can rewrite the expression as:

$$2^{2x} \cdot 2^{\frac{x}{3}} = 2^{2x + \frac{x}{3}}$$

Since $2x + \frac{x}{3} = \frac{7x}{3}$, we can rewrite the expression as:

$$2^{2x + \frac{x}{3}} = 2^{\frac{7x}{3}}$$

This expression is equivalent to the original expression $\sqrt[3]{128}$.

D. $\left(4\left(2^{\frac{1}{3}}\right)\right)^x$

This expression is equivalent to the original expression $\sqrt[3]{128}$. To see why, we can rewrite the expression as:

$$\left(4\left(2^{\frac{1}{3}}\right)\right)^x = \left(4 \cdot 2^{\frac{1}{3}}\right)^x$$

Using the property of exponents, we can rewrite the expression as:

$$\left(4 \cdot 2^{\frac{1}{3}}\right)^x = 4^x \cdot 2^{\frac{x}{3}}$$

Since $4 = 2^2$, we can rewrite the expression as:

$$4^x \cdot 2^{\frac{x}{3}} = (2^2)^x \cdot 2^{\frac{x}{3}}$$

Using the property of exponents, we can rewrite the expression as:

$$(2^2)^x \cdot 2^{\frac{x}{3}} = 2^{2x} \cdot 2^{\frac{x}{3}}$$

Using the property of exponents, we can rewrite the expression as:

$$2^{2x} \cdot 2^{\frac{x}{3}} = 2^{2x + \frac{x}{3}}$$

Since $2x + \frac{x}{3} = \frac{7x}{3}$, we can rewrite the expression as:

$$2^{2x + \frac{x}{3}} = 2^{\frac{7x}{3}}$$

This expression is equivalent to the original expression $\sqrt[3]{128}$.

Conclusion

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Q: What is the difference between a radical expression and an exponential expression?

A: A radical expression is an expression that contains a root, such as $\sqrt[3]{x}$, while an exponential expression is an expression that contains a power, such as $x^2$. While both types of expressions can be simplified, the rules for simplifying them are different.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to find the prime factorization of the number inside the radical sign. Then, you can rewrite the expression using the property of radicals, which states that $\sqrt[n]{a^n} = a$. For example, $\sqrt[3]{8} = \sqrt[3]{2^3} = 2$.

Q: What is the property of radicals?

A: The property of radicals states that $\sqrt[n]{a^n} = a$. This means that if you have a radical expression with a power of $n$, you can rewrite it as the $n$th root of the number inside the radical sign.

Q: How do I simplify an expression with multiple radicals?

A: To simplify an expression with multiple radicals, you need to simplify each radical separately and then combine the results. For example, $\sqrt[3]{8} + \sqrt[3]{27} = 2 + 3 = 5$.

Q: Can I simplify an expression with a negative number inside the radical sign?

A: Yes, you can simplify an expression with a negative number inside the radical sign. To do this, you need to rewrite the expression using the property of radicals, which states that $\sqrt[n]{-a^n} = -a$. For example, $\sqrt[3]{-8} = \sqrt[3]{-2^3} = -2$.

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

A: To simplify an expression with a fractional exponent, you need to rewrite the expression using the property of exponents, which states that $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, $x^{\frac{2}{3}} = \sqrt[3]{x^2}$.

Q: Can I simplify an expression with a radical sign and an exponential sign?

A: Yes, you can simplify an expression with a radical sign and an exponential sign. To do this, you need to rewrite the expression using the property of radicals and the property of exponents. For example, $\sqrt[3]{x^2} = x^{\frac{2}{3}}$.

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

A: To simplify an expression with a radical sign and a negative exponent, you need to rewrite the expression using the property of radicals and the property of exponents. For example, $\sqrt[3]{x^{-2}} = x^{-\frac{2}{3}}$.

Q: Can I simplify an expression with a radical sign and a variable inside the radical sign?

A: Yes, you can simplify an expression with a radical sign and a variable inside the radical sign. To do this, you need to rewrite the expression using the property of radicals and the property of exponents. For example, $\sqrt[3]{x^2} = x^{\frac{2}{3}}$.

Q: How do I simplify an expression with multiple radical signs and variables inside the radical signs?

A: To simplify an expression with multiple radical signs and variables inside the radical signs, you need to simplify each radical separately and then combine the results. For example, $\sqrt[3]{x^2} + \sqrt[3]{y^3} = x^{\frac{2}{3}} + y$.

Conclusion

In this article, we have answered some frequently asked questions about simplifying radical expressions. We have covered topics such as the difference between radical expressions and exponential expressions, the property of radicals, and how to simplify expressions with multiple radicals, negative numbers, fractional exponents, and variables inside the radical signs. We hope that this article has provided a helpful guide for readers.