Not breaking down the expression into its prime factors

Oct 13, 2025 by talynt.org 56 views

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

Understanding the Problem

In this problem, we are given the expression $$24^{\frac{1}{3}}$$ and asked to find an equivalent expression from the given options. To solve this problem, we need to understand the concept of exponentiation and radicals.

What is Exponentiation?

Exponentiation is a mathematical operation that involves raising a number to a power. In this case, we have $$24^{\frac{1}{3}}$$, which means we need to raise 24 to the power of $$\frac{1}{3}$$.

What is a Radical?

A radical is a mathematical expression that involves a root of a number. In this case, we have $$24^{\frac{1}{3}}$$, which can be written as $$\sqrt[3]{24}$$.

Breaking Down the Expression

To simplify the expression $$24^{\frac{1}{3}}$$, we can break it down into its prime factors. We can write 24 as $$2^3 \times 3$$.

Simplifying the Expression

Now that we have broken down 24 into its prime factors, we can simplify the expression $$24^{\frac{1}{3}}$$. We can write it as $$\sqrt[3]{2^3 \times 3}$$.

Using the Properties of Radicals

We can use the properties of radicals to simplify the expression further. We can write $$\sqrt[3]{2^3 \times 3}$$ as $$\sqrt[3]{2^3} \times \sqrt[3]{3}$$.

Simplifying the Cube Root

We can simplify the cube root of $$2^3$$ as $$2$$, since $$2^3 = 8$$ and $$\sqrt[3]{8} = 2$$.

Simplifying the Expression

Now that we have simplified the cube root of $$2^3$$, we can simplify the expression further. We can write $$2 \times \sqrt[3]{3}$$ as $$2\sqrt[3]{3}$$.

Conclusion

In conclusion, the expression $$24^{\frac{1}{3}}$$ is equivalent to $$2\sqrt[3]{3}$$.

Q&A

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 simplify the expression $$24^{\frac{1}{3}}$$?

A: To simplify the expression $$24^{\frac{1}{3}}$$, we can break it down into its prime factors and use the properties of radicals to simplify it further.

Q: What is the cube root of $$2^3$$?

A: The cube root of $$2^3$$ is $$2$$, since $$2^3 = 8$$ and $$\sqrt[3]{8} = 2$$.

Q: How do you simplify the cube root of $$2^3$$?

A: We can simplify the cube root of $$2^3$$ by writing it as $$\sqrt[3]{2^3}$$ and then simplifying it further.

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

A: The final simplified expression for $$24^{\frac{1}{3}}$$ is $$2\sqrt[3]{3}$$.

Common Mistakes

Not breaking down the expression into its prime factors: Failing to break down the expression into its prime factors can make it difficult to simplify the expression.
Not using the properties of radicals: Failing to use the properties of radicals can make it difficult to simplify the expression.
Not simplifying the cube root: Failing to simplify the cube root can make the expression more complicated than it needs to be.

Tips and Tricks

Use the properties of radicals to simplify the expression: Using the properties of radicals can make it easier to simplify the expression.
Break down the expression into its prime factors: Breaking down the expression into its prime factors can make it easier to simplify the expression.
Simplify the cube root: Simplifying the cube root can make the expression more straightforward.

Conclusion

In conclusion, the expression $$24^{\frac{1}{3}}$$ is equivalent to $$2\sqrt[3]{3}$$. We can simplify the expression by breaking it down into its prime factors and using the properties of radicals. By following these steps, we can simplify the expression and arrive at the final answer.