Understanding the Cube Root Formula

Oct 13, 2025 by magicalceylontours.com 109 views

What is the Cube Root of $$8x^{27}$$?

The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In mathematical terms, it is denoted by the symbol ∛. The cube root of a number can be calculated using various methods, including prime factorization and the use of a calculator.

Understanding the Cube Root Formula

The cube root formula is given by:

∛(a^3) = a

where 'a' is the number inside the cube root. This formula can be used to simplify cube roots of perfect cubes.

Breaking Down the Expression $$8x^{27}$$

To find the cube root of $$8x^{27}$$, we need to break down the expression into its prime factors. The expression can be written as:

8x^{27} = 2^3 \cdot x^{27}

Applying the Cube Root Formula

Now, we can apply the cube root formula to the expression:

∛($$8x^{27}$$) = ∛($$2^3 \cdot x^{27}$$)

Using the formula, we get:

∛($$8x^{27}$$) = ∛($$2^3$$) ∛($$x^{27}$$)

Simplifying the Cube Root

The cube root of $$2^3$$ is simply 2, since $$2^3 = 8$$ and ∛8 = 2.

The cube root of $$x^{27}$$ can be simplified using the rule:

∛($$x^n$$) = $$x^{n/3}$$

Applying this rule, we get:

∛($$x^{27}$$) = $$x^{27/3}$$

Simplifying further, we get:

∛($$x^{27}$$) = $$x^9$$

Combining the Results

Now, we can combine the results to get the final answer:

∛($$8x^{27}$$) = 2 ∛($$x^{27}$$)

Substituting the simplified expression for ∛($$x^{27}$$), we get:

∛($$8x^{27}$$) = 2 $$x^9$$

Conclusion

Therefore, the cube root of $$8x^{27}$$ is $$2x^9$$.

Comparison with the Options

Comparing our result with the options given, we can see that the correct answer is:

B. $$2x^9$$

The other options are incorrect, as they do not match the result we obtained using the cube root formula.

Key Takeaways

The cube root formula is given by ∛(a^3) = a.
To find the cube root of an expression, we need to break it down into its prime factors.
The cube root of a perfect cube can be simplified using the formula ∛(a^3) = a.
The cube root of $$x^n$$ can be simplified using the rule ∛($$x^n$$) = $$x^{n/3}$$.

By following these steps and using the cube root formula, we can simplify cube roots of complex expressions and arrive at the correct answer.
Cube Root Q&A: Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about cube roots, including their definition, properties, and how to simplify them.

Q: What is a cube root?

A: A cube root is a value that, when multiplied by itself twice, gives the original number. It is denoted by the symbol ∛.

Q: How do I find the cube root of a number?

A: To find the cube root of a number, you can use the cube root formula: ∛(a^3) = a. You can also use a calculator or a mathematical software to find the cube root.

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

A: A square root is a value that, when multiplied by itself, gives the original number. A cube root is a value that, when multiplied by itself twice, gives the original number.

Q: Can I simplify a cube root using algebraic manipulations?

A: Yes, you can simplify a cube root using algebraic manipulations. For example, if you have ∛($$x^3$$), you can simplify it to x.

Q: How do I simplify a cube root of a variable expression?

A: To simplify a cube root of a variable expression, you need to break it down into its prime factors. For example, if you have ∛($$x^3y3$$), you can simplify it to xy.

Q: Can I use the cube root formula to simplify a cube root of a fraction?

A: Yes, you can use the cube root formula to simplify a cube root of a fraction. For example, if you have ∛($$\frac{x^3}{y3}$$), you can simplify it to $$\frac{x}{y}$$.

Q: How do I find the cube root of a negative number?

A: To find the cube root of a negative number, you need to use the definition of the cube root. For example, if you have ∛(-8), you can find it by using the definition: ∛(-8) = -∛8.

Q: Can I use a calculator to find the cube root of a decimal number?

A: Yes, you can use a calculator to find the cube root of a decimal number. Most calculators have a cube root button that you can use to find the cube root of a decimal number.

Q: How do I find the cube root of a complex number?

A: To find the cube root of a complex number, you need to use the definition of the cube root. For example, if you have ∛(-8 + 6i), you can find it by using the definition: ∛(-8 + 6i) = -∛8 + ∛6i.

Q: Can I use the cube root formula to simplify a cube root of a polynomial?

A: Yes, you can use the cube root formula to simplify a cube root of a polynomial. For example, if you have ∛($$x^3 + 3x^2 + 2x + 1$$), you can simplify it by using the cube root formula.

Conclusion

In this article, we have addressed some of the most frequently asked questions about cube roots, including their definition, properties, and how to simplify them. We hope that this article has been helpful in clarifying any doubts you may have had about cube roots.

Key Takeaways

A cube root is a value that, when multiplied by itself twice, gives the original number.
To find the cube root of a number, you can use the cube root formula: ∛(a^3) = a.
You can simplify a cube root using algebraic manipulations.
You can use the cube root formula to simplify a cube root of a fraction.
You can use a calculator to find the cube root of a decimal number.
You can use the cube root formula to simplify a cube root of a polynomial.

By following these steps and using the cube root formula, you can simplify cube roots of complex expressions and arrive at the correct answer.