Which Expression Is Equivalent To $24^{\frac{1}{3}}$?A. 2 3 2 \sqrt{3} 2 3 ​ B. 2 3 3 2 \sqrt[3]{3} 2 3 3 ​ C. 2 6 2 \sqrt{6} 2 6 ​ D. 2 6 3 2 \sqrt[3]{6} 2 3 6 ​

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Understanding the Basics of Radical Expressions

Radical expressions are a fundamental concept in mathematics, and they play a crucial role in algebra, geometry, and other branches of mathematics. A radical expression is a mathematical expression that involves a root or a power of a number. In this article, we will focus on simplifying radical expressions and finding equivalent expressions.

What is a Radical Expression?

A radical expression is a mathematical expression that involves a root or a power of a number. It is denoted by a symbol called the radical sign, which is usually represented by a horizontal line above the root. For example, the expression $\sqrt{16}$ is a radical expression, where the root is 16 and the index is 2.

Types of Radical Expressions

There are two main types of radical expressions: square roots and cube roots. A square root is a radical expression that involves a root of 2, while a cube root is a radical expression that involves a root of 3.

Simplifying Radical Expressions

Simplifying radical expressions involves finding the simplest form of a radical expression. This can be done by factoring the radicand (the number inside the radical sign) and then simplifying the expression.

Example 1: Simplifying a Square Root

Let's consider the expression $\sqrt{16}.$ To simplify this expression, we need to find the factors of 16. The factors of 16 are 1, 2, 4, 8, and 16. Since 16 is a perfect square, we can simplify the expression as follows:

16=4×4=4×4=2×2=4\sqrt{16} = \sqrt{4 \times 4} = \sqrt{4} \times \sqrt{4} = 2 \times 2 = 4

Example 2: Simplifying a Cube Root

Let's consider the expression $\sqrt[3]{27}.$ To simplify this expression, we need to find the factors of 27. The factors of 27 are 1, 3, 9, and 27. Since 27 is a perfect cube, we can simplify the expression as follows:

273=3×3×33=33×33×33=3×3×3=27\sqrt[3]{27} = \sqrt[3]{3 \times 3 \times 3} = \sqrt[3]{3} \times \sqrt[3]{3} \times \sqrt[3]{3} = 3 \times 3 \times 3 = 27

Which Expression is Equivalent to $24^{\frac{1}{3}}$?

Now that we have a good understanding of simplifying radical expressions, let's consider the expression $24^{\frac{1}{3}}.$ To simplify this expression, we need to find the factors of 24. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Step 1: Factorize the Radicand

To simplify the expression $24^{\frac{1}{3}},$ we need to factorize the radicand (24). We can factorize 24 as follows:

24=2×2×2×324 = 2 \times 2 \times 2 \times 3

Step 2: Simplify the Expression

Now that we have factorized the radicand, we can simplify the expression as follows:

2413=(2×2×2×3)13=2×2×2×313=2×2×2×33=8×3324^{\frac{1}{3}} = (2 \times 2 \times 2 \times 3)^{\frac{1}{3}} = 2 \times 2 \times 2 \times 3^{\frac{1}{3}} = 2 \times 2 \times 2 \times \sqrt[3]{3} = 8 \times \sqrt[3]{3}

Which Expression is Equivalent to $24^{\frac{1}{3}}$?

Now that we have simplified the expression $24^{\frac{1}{3}},$ we can compare it with the given options. The options are:

A. 232 \sqrt{3} B. 2332 \sqrt[3]{3} C. 262 \sqrt{6} D. 2632 \sqrt[3]{6}

Comparing the Options

Let's compare the simplified expression $8 \times \sqrt[3]{3}$ with the given options. We can see that option B is equivalent to the simplified expression.

Conclusion

In this article, we have discussed the basics of radical expressions and how to simplify them. We have also considered the expression $24^{\frac{1}{3}}$ and simplified it to $8 \times \sqrt[3]{3}.$ We have compared the simplified expression with the given options and found that option B is equivalent to the simplified expression.

Final Answer

The final answer is:

B. 2332 \sqrt[3]{3}

Understanding the Basics of Radical Expressions

Radical expressions are a fundamental concept in mathematics, and they play a crucial role in algebra, geometry, and other branches of mathematics. A radical expression is a mathematical expression that involves a root or a power of a number. In this article, we will focus on simplifying radical expressions and finding equivalent expressions.

What is a Radical Expression?

A radical expression is a mathematical expression that involves a root or a power of a number. It is denoted by a symbol called the radical sign, which is usually represented by a horizontal line above the root. For example, the expression $\sqrt{16}$ is a radical expression, where the root is 16 and the index is 2.

Types of Radical Expressions

There are two main types of radical expressions: square roots and cube roots. A square root is a radical expression that involves a root of 2, while a cube root is a radical expression that involves a root of 3.

Simplifying Radical Expressions

Simplifying radical expressions involves finding the simplest form of a radical expression. This can be done by factoring the radicand (the number inside the radical sign) and then simplifying the expression.

Example 1: Simplifying a Square Root

Let's consider the expression $\sqrt{16}.$ To simplify this expression, we need to find the factors of 16. The factors of 16 are 1, 2, 4, 8, and 16. Since 16 is a perfect square, we can simplify the expression as follows:

16=4×4=4×4=2×2=4\sqrt{16} = \sqrt{4 \times 4} = \sqrt{4} \times \sqrt{4} = 2 \times 2 = 4

Example 2: Simplifying a Cube Root

Let's consider the expression $\sqrt[3]{27}.$ To simplify this expression, we need to find the factors of 27. The factors of 27 are 1, 3, 9, and 27. Since 27 is a perfect cube, we can simplify the expression as follows:

273=3×3×33=33×33×33=3×3×3=27\sqrt[3]{27} = \sqrt[3]{3 \times 3 \times 3} = \sqrt[3]{3} \times \sqrt[3]{3} \times \sqrt[3]{3} = 3 \times 3 \times 3 = 27

Which Expression is Equivalent to $24^{\frac{1}{3}}$?

Now that we have a good understanding of simplifying radical expressions, let's consider the expression $24^{\frac{1}{3}}.$ To simplify this expression, we need to find the factors of 24. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Step 1: Factorize the Radicand

To simplify the expression $24^{\frac{1}{3}},$ we need to factorize the radicand (24). We can factorize 24 as follows:

24=2×2×2×324 = 2 \times 2 \times 2 \times 3

Step 2: Simplify the Expression

Now that we have factorized the radicand, we can simplify the expression as follows:

2413=(2×2×2×3)13=2×2×2×313=2×2×2×33=8×3324^{\frac{1}{3}} = (2 \times 2 \times 2 \times 3)^{\frac{1}{3}} = 2 \times 2 \times 2 \times 3^{\frac{1}{3}} = 2 \times 2 \times 2 \times \sqrt[3]{3} = 8 \times \sqrt[3]{3}

Which Expression is Equivalent to $24^{\frac{1}{3}}$?

Now that we have simplified the expression $24^{\frac{1}{3}},$ we can compare it with the given options. The options are:

A. 232 \sqrt{3} B. 2332 \sqrt[3]{3} C. 262 \sqrt{6} D. 2632 \sqrt[3]{6}

Comparing the Options

Let's compare the simplified expression $8 \times \sqrt[3]{3}$ with the given options. We can see that option B is equivalent to the simplified expression.

Conclusion

In this article, we have discussed the basics of radical expressions and how to simplify them. We have also considered the expression $24^{\frac{1}{3}}$ and simplified it to $8 \times \sqrt[3]{3}.$ We have compared the simplified expression with the given options and found that option B is equivalent to the simplified expression.

Final Answer

The final answer is:

B. 2332 \sqrt[3]{3}

Q&A: Simplifying Radical Expressions

Q: What is a radical expression?

A: A radical expression is a mathematical expression that involves a root or a power of a number.

Q: What are the two main types of radical expressions?

A: The two main types of radical expressions are square roots and cube roots.

Q: How do you simplify a radical expression?

A: To simplify a radical expression, you need to factorize the radicand (the number inside the radical sign) and then simplify the expression.

Q: What is the difference between a square root and a cube root?

A: A square root is a radical expression that involves a root of 2, while a cube root is a radical expression that involves a root of 3.

Q: How do you simplify a cube root?

A: To simplify a cube root, you need to find the factors of the number inside the cube root sign and then simplify the expression.

Q: What is the final answer to the expression $24^{\frac{1}{3}}$?

A: The final answer to the expression $24^{\frac{1}{3}}$ is $8 \times \sqrt[3]{3}$, which is equivalent to option B.

Q: What is the equivalent expression to $24^{\frac{1}{3}}$?

A: The equivalent expression to $24^{\frac{1}{3}}$ is $2 \sqrt[3]{3}$.

Q: How do you compare the simplified expression with the given options?

A: To compare the simplified expression with the given options, you need to look for the option that is equivalent to the simplified expression.

Q: What is the final answer to the question "Which expression is equivalent to $24^{\frac{1}{3}}$?"

A: The final answer to the question "Which expression is equivalent to $24^{\frac{1}{3}}$?" is option B, which is $2 \sqrt[3]{3}$.