Simplify The Radical 27 A 9 B 12 C 6 3 \sqrt[3]{27 A^9 B^{12} C^6} 3 27 A 9 B 12 C 6 .A. 3 A 6 B 9 C 3 3 A^6 B^9 C^3 3 A 6 B 9 C 3 B. 9 A 6 B 9 C 3 9 A^6 B^9 C^3 9 A 6 B 9 C 3 C. 3 A 3 B 4 C 2 3 A^3 B^4 C^2 3 A 3 B 4 C 2 D. 9 A 3 B 4 C 2 9 A^3 B^4 C^2 9 A 3 B 4 C 2

Mar 12, 2025 by ADMIN 268 views

$Simplify the Radical $\sqrt[3]{27 a^9 b^{12} c^6}$$

Understanding the Problem

When dealing with radicals, it's essential to understand the properties of exponents and how they interact with the radical sign. In this problem, we're given the expression $\sqrt[3]{27 a^9 b^{12} c^6}$ and asked to simplify it. To simplify the radical, we need to break down the expression into its prime factors and then use the properties of exponents to simplify the expression.

Breaking Down the Expression

Let's start by breaking down the expression into its prime factors. We can write $27$ as $3^3$ , so the expression becomes:

\sqrt[3]{3^3 a^9 b^{12} c^6}

Using the Properties of Exponents

Now that we have the expression broken down into its prime factors, we can use the properties of exponents to simplify the expression. When we take the cube root of a number, we are essentially raising that number to the power of $\frac{1}{3}$ . So, we can rewrite the expression as:

3^{\frac{3}{3}} a^{\frac{9}{3}} b^{\frac{12}{3}} c^{\frac{6}{3}}

Simplifying the Exponents

Now that we have the expression rewritten with the exponents, we can simplify them. When we divide the exponent by the cube root, we get:

3^1 a^3 b^4 c^2

Comparing the Options

Now that we have simplified the radical, we can compare our answer to the options given. The correct answer is:

3 a^3 b^4 c^2

This is option C.

Conclusion

In this problem, we were given the expression $\sqrt[3]{27 a^9 b^{12} c^6}$ and asked to simplify it. We broke down the expression into its prime factors, used the properties of exponents to simplify the expression, and then compared our answer to the options given. The correct answer is option C, which is $3 a^3 b^4 c^2$ .

Tips and Tricks

When dealing with radicals, it's essential to understand the properties of exponents and how they interact with the radical sign.
Breaking down the expression into its prime factors can help simplify the radical.
Using the properties of exponents can help simplify the expression.
Comparing the answer to the options given can help determine the correct answer.

Common Mistakes

Failing to break down the expression into its prime factors can make it difficult to simplify the radical.
Not using the properties of exponents can make it difficult to simplify the expression.
Not comparing the answer to the options given can lead to an incorrect answer.

Real-World Applications

Simplifying radicals is an essential skill in mathematics, particularly in algebra and geometry.
Understanding the properties of exponents and how they interact with the radical sign is crucial in many real-world applications, such as physics and engineering.

Practice Problems

Simplify the radical $\sqrt[3]{64 a^6 b^{18} c^9}$ .
Simplify the radical $\sqrt[3]{125 a^{12} b^{24} c^{18}}$ .

Solutions

$\sqrt[3]{64 a^6 b^{18} c^9} = 4 a^2 b^6 c^3$
$\sqrt[3]{125 a^{12} b^{24} c^{18}} = 5 a^4 b^8 c^6$

Conclusion

Simplifying radicals is an essential skill in mathematics, particularly in algebra and geometry. Understanding the properties of exponents and how they interact with the radical sign is crucial in many real-world applications. By breaking down the expression into its prime factors, using the properties of exponents, and comparing the answer to the options given, we can simplify radicals and determine the correct answer.

Frequently Asked Questions

Q: What is the first step in simplifying a radical?

A: The first step in simplifying a radical is to break down the expression into its prime factors.

Q: How do I break down an expression into its prime factors?

A: To break down an expression into its prime factors, you need to identify the prime numbers that multiply together to give the original expression. For example, if you have the expression $27 a^9 b^{12} c^6$ , you can break it down into its prime factors as $3^3 a^9 b^{12} c^6$ .

Q: What is the next step in simplifying a radical?

A: The next step in simplifying a radical is to use the properties of exponents to simplify the expression.

Q: How do I use the properties of exponents to simplify a radical?

A: To use the properties of exponents to simplify a radical, you need to raise each factor to the power of $\frac{1}{3}$ . For example, if you have the expression $3^3 a^9 b^{12} c^6$ , you can raise each factor to the power of $\frac{1}{3}$ to get $3^{\frac{3}{3}} a^{\frac{9}{3}} b^{\frac{12}{3}} c^{\frac{6}{3}}$ .

Q: How do I simplify the exponents in a radical?

A: To simplify the exponents in a radical, you need to divide each exponent by the cube root. For example, if you have the expression $3^{\frac{3}{3}} a^{\frac{9}{3}} b^{\frac{12}{3}} c^{\frac{6}{3}}$ , you can simplify the exponents by dividing each one by 3 to get $3^1 a^3 b^4 c^2$ .

Q: How do I compare my answer to the options given?

A: To compare your answer to the options given, you need to look at the simplified expression and match it to one of the options. For example, if you have the expression $3^1 a^3 b^4 c^2$ , you can compare it to the options given to see that it matches option C.

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

A: Some common mistakes to avoid when simplifying radicals include failing to break down the expression into its prime factors, not using the properties of exponents, and not comparing the answer to the options given.

Q: What are some real-world applications of simplifying radicals?

A: Simplifying radicals has many real-world applications, including physics and engineering. Understanding the properties of exponents and how they interact with the radical sign is crucial in many of these applications.

Q: How can I practice simplifying radicals?

A: You can practice simplifying radicals by working through practice problems, such as simplifying the radical $\sqrt[3]{64 a^6 b^{18} c^9}$ or $\sqrt[3]{125 a^{12} b^{24} c^{18}}$ .

Solutions

$\sqrt[3]{64 a^6 b^{18} c^9} = 4 a^2 b^6 c^3$
$\sqrt[3]{125 a^{12} b^{24} c^{18}} = 5 a^4 b^8 c^6$