Evaluate The Expression By Hand, If Possible. Variables Represent Any Real Number.$\sqrt[3]{-27 X^3}$Select The Correct Choice Below And, If Necessary, Fill In The Answer Box To Complete Your Choice.A. $\sqrt[3]{-27 X^3} =$

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Understanding the Problem

The given expression is $\sqrt[3]{-27 x^3}$, and we are asked to evaluate it by hand, if possible. Variables represent any real number, which means we can use any real number value for $x$.

Breaking Down the Expression

To evaluate the expression, we need to understand the properties of cube roots and exponents. The cube root of a number is a value that, when multiplied by itself twice, gives the original number. In this case, we have $\sqrt[3]{-27 x^3}$, which can be rewritten as $(-27 x^3)^{1/3}$.

Simplifying the Expression

Using the property of exponents that $(a^m)^n = a^{m \cdot n}$, we can simplify the expression as follows:

$$(-27 x^3)^{1/3} = (-27)^{1/3} \cdot (x^3)^{1/3}$$

Evaluating the Cube Root of -27

The cube root of -27 is a value that, when multiplied by itself twice, gives -27. Since -27 is a perfect cube (-27 = -3^3), we can easily find its cube root:

$$(-27)^{1/3} = -3$$

Evaluating the Cube Root of $x^3$

The cube root of $x^3$ is simply $x$, since $x^3$ is a perfect cube:

$$(x^3)^{1/3} = x$$

Combining the Results

Now that we have evaluated the cube root of -27 and $x^3$, we can combine the results to get the final answer:

$$(-27 x^3)^{1/3} = (-27)^{1/3} \cdot (x^3)^{1/3} = -3 \cdot x = \boxed{-3x}$$

Conclusion

In conclusion, the expression $\sqrt[3]{-27 x^3}$ can be evaluated by hand as $-3x$. This is because the cube root of -27 is -3, and the cube root of $x^3$ is $x$. By combining these results, we get the final answer of $-3x$.

Example Use Case

Suppose we want to evaluate the expression $\sqrt[3]{-27 x^3}$ when $x = 2$. We can simply substitute $x = 2$ into the expression and evaluate it:

$$\sqrt[3]{-27 (2)^3} = \sqrt[3]{-27 \cdot 8} = \sqrt[3]{-216} = -6$$

Tips and Variations

• When evaluating expressions with cube roots, it's often helpful to use the property of exponents that $(a^m)^n = a^{m \cdot n}$.
• If the expression involves a variable, be sure to substitute the variable with a specific value before evaluating the expression.
• Cube roots can be used to solve equations involving exponents. For example, if we have the equation $x^3 = 27$, we can take the cube root of both sides to get $x = \sqrt[3]{27} = 3$.

Common Mistakes

• When evaluating expressions with cube roots, it's easy to get confused between the cube root and the square root. Remember that the cube root of a number is a value that, when multiplied by itself twice, gives the original number.
• When simplifying expressions with cube roots, be sure to use the property of exponents that $(a^m)^n = a^{m \cdot n}$.
• When evaluating expressions with variables, be sure to substitute the variable with a specific value before evaluating the expression.

Real-World Applications

• Cube roots are used in many real-world applications, such as physics and engineering. For example, the cube root of a number can be used to calculate the volume of a cube.
• Cube roots are also used in finance to calculate the return on investment (ROI) of a stock or bond.
• In computer science, cube roots are used in algorithms for solving equations involving exponents.

Conclusion

In conclusion, evaluating the expression $\sqrt[3]{-27 x^3}$ by hand is a straightforward process that involves understanding the properties of cube roots and exponents. By breaking down the expression and simplifying it step by step, we can arrive at the final answer of $-3x$. This is a useful skill to have in mathematics and can be applied to a variety of real-world problems.

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Frequently Asked Questions

Q: What is the cube root of -27?

A: The cube root of -27 is -3, since -27 is a perfect cube (-27 = -3^3).

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

A: The cube root of $x^3$ is simply $x$, since $x^3$ is a perfect cube.

Q: How do I evaluate the expression $\sqrt[3]{-27 x^3}$?

A: To evaluate the expression, you can break it down into two parts: the cube root of -27 and the cube root of $x^3$. Then, you can combine the results to get the final answer.

Q: What is the final answer to the expression $\sqrt[3]{-27 x^3}$?

A: The final answer to the expression $\sqrt[3]{-27 x^3}$ is $-3x$.

Q: Can I use the expression $\sqrt[3]{-27 x^3}$ to solve equations involving exponents?

A: Yes, you can use the expression $\sqrt[3]{-27 x^3}$ to solve equations involving exponents. For example, if you have the equation $x^3 = 27$, you can take the cube root of both sides to get $x = \sqrt[3]{27} = 3$.

Q: What are some common mistakes to avoid when evaluating expressions with cube roots?

A: Some common mistakes to avoid when evaluating expressions with cube roots include:

• Getting confused between the cube root and the square root
• Not using the property of exponents that $(a^m)^n = a^{m \cdot n}$
• Not substituting variables with specific values before evaluating the expression

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

A: Some real-world applications of cube roots include:

• Calculating the volume of a cube
• Calculating the return on investment (ROI) of a stock or bond
• Solving equations involving exponents in physics and engineering

Q: Can I use cube roots to solve equations involving variables?

A: Yes, you can use cube roots to solve equations involving variables. For example, if you have the equation $x^3 = 27$, you can take the cube root of both sides to get $x = \sqrt[3]{27} = 3$.

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

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

Q: How do I simplify expressions with cube roots?

A: To simplify expressions with cube roots, you can use the property of exponents that $(a^m)^n = a^{m \cdot n}$. You can also break down the expression into two parts: the cube root of the coefficient and the cube root of the variable.

Q: Can I use cube roots to solve equations involving fractions?

A: Yes, you can use cube roots to solve equations involving fractions. For example, if you have the equation $\frac{x^3}{27} = 1$, you can take the cube root of both sides to get $\frac{x}{3} = 1$.

Q: What are some tips for evaluating expressions with cube roots?

A: Some tips for evaluating expressions with cube roots include:

• Breaking down the expression into two parts: the cube root of the coefficient and the cube root of the variable
• Using the property of exponents that $(a^m)^n = a^{m \cdot n}$
• Substituting variables with specific values before evaluating the expression

Q: Can I use cube roots to solve equations involving decimals?

A: Yes, you can use cube roots to solve equations involving decimals. For example, if you have the equation $x^3 = 27.5$, you can take the cube root of both sides to get $x = \sqrt[3]{27.5}$.