What Is 27 X 9 3 \sqrt[3]{27 X^9} 3 27 X 9 ​ ?A. 3 X 6 3 X^6 3 X 6 B. 3 X 3 3 X^3 3 X 3 C. 9 X 3 9 X^3 9 X 3 D. 9 X 6 9 X^6 9 X 6

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Understanding the Problem

To solve the problem, we need to understand the concept of cube roots and how they apply to algebraic expressions. The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In mathematical terms, $\sqrt[3]{a} = b$ means that $b^3 = a$. We will use this concept to simplify the given expression.

Breaking Down the Expression

The given expression is $\sqrt[3]{27 x^9}$. To simplify this expression, we need to break it down into its prime factors. The number 27 can be expressed as $3^3$, and $x^9$ can be expressed as $(x^3)^3$. Therefore, the expression can be rewritten as $\sqrt[3]{(3^3) (x^3)^3}$.

Simplifying the Expression

Now that we have broken down the expression into its prime factors, we can simplify it by applying the properties of cube roots. When we take the cube root of a product, we can take the cube root of each factor separately. Therefore, we can rewrite the expression as $\sqrt[3]{3^3} \sqrt[3]{(x^3)^3}$.

Evaluating the Cube Roots

Now that we have separated the factors, we can evaluate the cube roots. The cube root of $3^3$ is simply 3, and the cube root of $(x^3)^3$ is $x^3$. Therefore, the expression simplifies to $3 x^3$.

Conclusion

In conclusion, the value of $\sqrt[3]{27 x^9}$ is $3 x^3$. This is the correct answer among the given options.

Understanding the Options

Let's take a closer look at the options provided:

• A. $3 x^6$
• B. $3 x^3$
• C. $9 x^3$
• D. $9 x^6$

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options. Option A is $3 x^6$, which is not equal to $3 x^3$. Option C is $9 x^3$, which is also not equal to $3 x^3$. Option D is $9 x^6$, which is not equal to $3 x^3$. Only option B is $3 x^3$, which is equal to the simplified expression.

Final Answer

The final answer is option B, $3 x^3$.

Why is this Important?

Understanding how to simplify cube roots is an important concept in algebra. It can be used to solve a wide range of problems, from simple expressions to complex equations. By mastering this concept, you can become more confident and proficient in solving mathematical problems.

Real-World Applications

The concept of cube roots has many real-world applications. For example, in physics, the cube root is used to calculate the volume of a cube. In engineering, the cube root is used to calculate the stress on a material. In finance, the cube root is used to calculate the return on investment.

Common Mistakes

When simplifying cube roots, there are several common mistakes to avoid. One mistake is to forget to separate the factors. Another mistake is to forget to evaluate the cube roots. By being aware of these common mistakes, you can avoid them and ensure that your answers are correct.

Tips and Tricks

Here are some tips and tricks to help you simplify cube roots:

• Always break down the expression into its prime factors.
• Always separate the factors when taking the cube root of a product.
• Always evaluate the cube roots separately.
• Always check your work to ensure that your answer is correct.

Conclusion

In conclusion, the value of $\sqrt[3]{27 x^9}$ is $3 x^3$. This is the correct answer among the given options. By understanding how to simplify cube roots, you can become more confident and proficient in solving mathematical problems. Remember to always break down the expression into its prime factors, separate the factors when taking the cube root of a product, evaluate the cube roots separately, and check your work to ensure that your answer is correct.

Q: What is a cube root?

A: A cube root is a value that, when multiplied by itself three times, gives the original number. In mathematical terms, $\sqrt[3]{a} = b$ means that $b^3 = a$.

Q: How do I simplify a cube root?

A: To simplify a cube root, you need to break down the expression into its prime factors and separate the factors when taking the cube root of a product. Then, evaluate the cube roots separately.

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 three times, gives the original number.

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

A: To calculate the cube root of a number, you can use a calculator or a mathematical formula. The formula for calculating the cube root of a number is $\sqrt[3]{a} = \frac{a^\frac{1}{3}}{1}$.

Q: What are some real-world applications of cube roots?

A: Cube roots have many real-world applications, including physics, engineering, and finance. In physics, the cube root is used to calculate the volume of a cube. In engineering, the cube root is used to calculate the stress on a material. In finance, the cube root is used to calculate the return on investment.

Q: What are some common mistakes to avoid when simplifying cube roots?

A: Some common mistakes to avoid when simplifying cube roots include forgetting to separate the factors, forgetting to evaluate the cube roots, and not checking your work.

Q: How do I check my work when simplifying cube roots?

A: To check your work when simplifying cube roots, you need to plug your answer back into the original expression and verify that it is true.

Q: What are some tips and tricks for simplifying cube roots?

A: Some tips and tricks for simplifying cube roots include breaking down the expression into its prime factors, separating the factors when taking the cube root of a product, evaluating the cube roots separately, and checking your work.

Q: Can I use a calculator to simplify cube roots?

A: Yes, you can use a calculator to simplify cube roots. However, it's also important to understand the mathematical concept behind the calculation.

Q: How do I simplify a cube root with a variable?

A: To simplify a cube root with a variable, you need to break down the expression into its prime factors and separate the factors when taking the cube root of a product. Then, evaluate the cube roots separately.

Q: What is the difference between a cube root and a power of 3?

A: A cube root is a value that, when multiplied by itself three times, gives the original number. A power of 3 is a value that is raised to the power of 3.

Q: How do I simplify a cube root with a negative number?

A: To simplify a cube root with a negative number, you need to break down the expression into its prime factors and separate the factors when taking the cube root of a product. Then, evaluate the cube roots separately.

Q: Can I simplify a cube root with a fraction?

A: Yes, you can simplify a cube root with a fraction. To do this, you need to break down the fraction into its prime factors and separate the factors when taking the cube root of a product. Then, evaluate the cube roots separately.

Q: How do I simplify a cube root with a decimal?

A: To simplify a cube root with a decimal, you need to break down the decimal into its prime factors and separate the factors when taking the cube root of a product. Then, evaluate the cube roots separately.

Q: What are some advanced topics related to cube roots?

A: Some advanced topics related to cube roots include cube roots of negative numbers, cube roots of fractions, and cube roots of decimals.