Select The Correct Answer.Which Expression Is Equivalent To $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$?A. 12 B. $30 \sqrt[3]{15}$ C. $ 12 5 3 12 \sqrt[3]{5} 12 3 5 [/tex] D. $12 \sqrt[3]{15}$

Mar 10, 2025 by ADMIN 195 views

**Simplifying Radical Expressions: A Step-by-Step Guide**

Understanding the Problem

When dealing with radical expressions, it's essential to understand the properties of radicals and how to simplify them. In this article, we'll explore how to simplify the given expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ and determine which of the provided options is equivalent to it.

The Basics of Radicals

A radical is a mathematical expression that involves a root, such as a square root or cube root. The cube root of a number, denoted by $\sqrt[3]{x}$, is a number that, when multiplied by itself twice, gives the original number. For example, $\sqrt[3]{27} = 3$ because $3 \times 3 \times 3 = 27$.

Simplifying the Given Expression

To simplify the given expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$, we can start by factoring out the common term $\sqrt[3]{15}$. This will give us:

21 \sqrt[3]{15} - 9 \sqrt[3]{15} = (21 - 9) \sqrt[3]{15}

Evaluating the Expression

Now, we can evaluate the expression inside the parentheses:

(21 - 9) \sqrt[3]{15} = 12 \sqrt[3]{15}

Comparing with the Options

Let's compare our simplified expression $12 \sqrt[3]{15}$ with the provided options:

A. 12: This option is incorrect because it doesn't include the radical term $\sqrt[3]{15}$.
B. $30 \sqrt[3]{15}$: This option is incorrect because it includes a different coefficient (30) and doesn't match our simplified expression.
C. $12 \sqrt[3]{5}$: This option is incorrect because it includes a different radicand (5) and doesn't match our simplified expression.
D. $12 \sqrt[3]{15}$: This option matches our simplified expression exactly.

Conclusion

In conclusion, the correct answer is D. $12 \sqrt[3]{15}$. This option matches our simplified expression exactly, and it's the only option that includes the correct coefficient and radicand.

Tips and Tricks

When dealing with radical expressions, it's essential to remember the following tips and tricks:

Always factor out common terms to simplify the expression.
Evaluate the expression inside the parentheses carefully.
Compare the simplified expression with the provided options carefully.
Make sure to include the correct coefficient and radicand in the final answer.

Practice Problems

If you want to practice simplifying radical expressions, try the following problems:

Simplify the expression $24 \sqrt[3]{32} - 12 \sqrt[3]{32}$.
Simplify the expression $18 \sqrt[3]{27} + 9 \sqrt[3]{27}$.
Simplify the expression $30 \sqrt[3]{125} - 15 \sqrt[3]{125}$.

Real-World Applications

Radical expressions have many real-world applications, such as:

Calculating volumes of irregular shapes.
Determining the length of shadows or diagonals.
Finding the area of complex shapes.

Final Thoughts

In conclusion, simplifying radical expressions is an essential skill in mathematics. By following the steps outlined in this article, you can simplify complex expressions and determine the correct answer. Remember to factor out common terms, evaluate the expression inside the parentheses carefully, and compare the simplified expression with the provided options carefully. With practice and patience, you'll become proficient in simplifying radical expressions and applying them to real-world problems.

Q: What is the difference between a radical and an exponent?

A: A radical is a mathematical expression that involves a root, such as a square root or cube root. An exponent, on the other hand, is a small number that is raised to a power, such as 2^3 or 3^4. While both radicals and exponents involve powers, they are used in different contexts and have different properties.

Q: How do I simplify a radical expression with multiple terms?

A: To simplify a radical expression with multiple terms, you can factor out the common term from each term. For example, if you have the expression $3 \sqrt[3]{12} + 2 \sqrt[3]{12}$, you can factor out the common term $\sqrt[3]{12}$ to get $(3 + 2) \sqrt[3]{12} = 5 \sqrt[3]{12}$.

Q: Can I simplify a radical expression by combining like terms?

A: Yes, you can simplify a radical expression by combining like terms. For example, if you have the expression $\sqrt[3]{16} + \sqrt[3]{16}$, you can combine the like terms to get $2 \sqrt[3]{16}$.

Q: How do I simplify a radical expression with a fraction?

A: To simplify a radical expression with a fraction, you can multiply the numerator and denominator by the radical. For example, if you have the expression $\frac{\sqrt[3]{27}}{3}$, you can multiply the numerator and denominator by $\sqrt[3]{27}$ to get $\frac{27}{3 \sqrt[3]{27}} = \frac{9}{\sqrt[3]{27}}$.

Q: Can I simplify a radical expression by canceling out common factors?

A: Yes, you can simplify a radical expression by canceling out common factors. For example, if you have the expression $\sqrt[3]{24} \times \sqrt[3]{6}$, you can cancel out the common factor $\sqrt[3]{6}$ to get $\sqrt[3]{24}$.

Q: How do I simplify a radical expression with a negative exponent?

A: To simplify a radical expression with a negative exponent, you can rewrite the expression with a positive exponent and then simplify. For example, if you have the expression $\sqrt[3]{x}^{-2}$, you can rewrite it as $\frac{1}{\sqrt[3]{x}^2}$ and then simplify.

Q: Can I simplify a radical expression by using the product rule?

A: Yes, you can simplify a radical expression by using the product rule. The product rule states that $\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{ab}$. For example, if you have the expression $\sqrt[3]{12} \times \sqrt[3]{15}$, you can use the product rule to get $\sqrt[3]{12 \times 15} = \sqrt[3]{180}$.

Q: How do I simplify a radical expression with a rational exponent?

A: To simplify a radical expression with a rational exponent, you can rewrite the expression with a radical and then simplify. For example, if you have the expression $x^{\frac{2}{3}}$, you can rewrite it as $\sqrt[3]{x^2}$ and then simplify.

Q: Can I simplify a radical expression by using the quotient rule?

A: Yes, you can simplify a radical expression by using the quotient rule. The quotient rule states that $\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\frac{a}{b}}$. For example, if you have the expression $\frac{\sqrt[3]{24}}{\sqrt[3]{6}}$, you can use the quotient rule to get $\sqrt[3]{\frac{24}{6}} = \sqrt[3]{4}$.

Q: How do I simplify a radical expression with a complex number?

A: To simplify a radical expression with a complex number, you can use the properties of radicals and complex numbers. For example, if you have the expression $\sqrt[3]{-8}$, you can rewrite it as $\sqrt[3]{-2^3} = -2$.

Q: Can I simplify a radical expression by using the conjugate?

A: Yes, you can simplify a radical expression by using the conjugate. The conjugate of a complex number is another complex number that has the same real part and the opposite imaginary part. For example, if you have the expression $\sqrt[3]{-1 + \sqrt{3}i}$, you can use the conjugate to simplify it.

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you can use the properties of radicals and variables. For example, if you have the expression $\sqrt[3]{x^2}$, you can simplify it by rewriting it as $x^{\frac{2}{3}}$.

Q: Can I simplify a radical expression by using the identity?

A: Yes, you can simplify a radical expression by using the identity. The identity states that $\sqrt[3]{a^3} = a$. For example, if you have the expression $\sqrt[3]{27}$, you can use the identity to simplify it to $3$.

Q: How do I simplify a radical expression with a negative number?

A: To simplify a radical expression with a negative number, you can use the properties of radicals and negative numbers. For example, if you have the expression $\sqrt[3]{-8}$, you can simplify it by rewriting it as $-2$.

Q: Can I simplify a radical expression by using the absolute value?

A: Yes, you can simplify a radical expression by using the absolute value. The absolute value of a number is its distance from zero, without considering direction. For example, if you have the expression $\sqrt[3]{-8}$, you can use the absolute value to simplify it to $2$.