Which Expression Is Equivalent To $\sqrt[4]{\frac{81}{16} A^8 B^{12} C^{16}}$?A. $\frac{3}{2} A^2\left|b^3\right| C^4$B. $\frac{3}{2} A^4 B^8 C^{12}$C. $\frac{9}{4} A^2\left|b^3\right| C^4$D. $\frac{9}{4} A^4 B^6

Feb 27, 2025 by ADMIN 213 views

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

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given problem: $\sqrt[4]{\frac{81}{16} a^8 b^{12} c^{16}}$ . We will break down the solution step by step, using the properties of exponents and radicals to arrive at the final answer.

Understanding the Problem

The given expression is a radical expression, which involves the fourth root of a fraction multiplied by variables raised to certain powers. Our goal is to simplify this expression and find an equivalent form.

Step 1: Simplify the Fraction

The first step is to simplify the fraction inside the radical expression. We can do this by finding the greatest common divisor (GCD) of the numerator and denominator.

\frac{81}{16} = \frac{3^4}{2^4}

Now, we can rewrite the fraction as:

\frac{3^4}{2^4} = \left(\frac{3}{2}\right)^4

Step 2: Apply the Power Rule

The next step is to apply the power rule, which states that for any non-negative integer $n$ , $(ab)^n = a^nb^n$ . We can use this rule to simplify the variables inside the radical expression.

a^8 = (a^4)^2
b^{12} = (b^6)^2
c^{16} = (c^8)^2

Now, we can rewrite the expression as:

\sqrt[4]{\left(\frac{3}{2}\right)^4 a^8 b^{12} c^{16}} = \sqrt[4]{\left(\frac{3}{2}\right)^4 (a^4)^2 (b^6)^2 (c^8)^2}

Step 3: Simplify the Radicals

The next step is to simplify the radicals using the property $\sqrt[n]{a^n} = a$ . We can apply this property to each of the variables inside the radical expression.

\sqrt[4]{\left(\frac{3}{2}\right)^4 (a^4)^2 (b^6)^2 (c^8)^2} = \left(\frac{3}{2}\right)^1 a^2 b^3 c^4

Step 4: Simplify the Absolute Value

The final step is to simplify the absolute value expression. We can do this by removing the absolute value sign and keeping the expression inside the same.

\left|b^3\right| = b^3

Conclusion

In conclusion, the simplified expression is $\frac{3}{2} a^2 b^3 c^4$ . This is the equivalent form of the given radical expression.

Answer

The correct answer is:

C. $\frac{9}{4} a^2\left|b^3\right| c^4$

Note that the answer is not among the options provided in the problem. However, we can rewrite the answer as:

C. $\frac{9}{4} a^2 b^3 c^4$

This is the equivalent form of the given radical expression.

Discussion

The given problem involves simplifying a radical expression using the properties of exponents and radicals. The solution requires a step-by-step approach, starting with simplifying the fraction, applying the power rule, simplifying the radicals, and finally simplifying the absolute value expression.

Key Takeaways

Simplifying radical expressions involves applying the properties of exponents and radicals.
The power rule states that for any non-negative integer $n$ , $(ab)^n = a^nb^n$ .
The property $\sqrt[n]{a^n} = a$ can be used to simplify radicals.
Absolute value expressions can be simplified by removing the absolute value sign and keeping the expression inside the same.

Practice Problems

Simplify the radical expression: $\sqrt[3]{\frac{64}{27} a^9 b^{18} c^{27}}$ .
Simplify the radical expression: $\sqrt[5]{\frac{100}{32} a^{15} b^{30} c^{40}}$ .
Simplify the radical expression: $\sqrt[4]{\frac{81}{16} a^8 b^{12} c^{16}}$ .

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "Calculus" by Michael Spivak

Conclusion

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In our previous article, we explored the process of simplifying radical expressions, with a focus on the given problem: $\sqrt[4]{\frac{81}{16} a^8 b^{12} c^{16}}$ . In this article, we will provide a Q&A guide to help you better understand the concept of simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that involves a root or a power of a number. It is denoted by the symbol $\sqrt[n]{a}$ , where $n$ is the index of the root and $a$ is the radicand.

Q: What are the properties of exponents and radicals?

A: The properties of exponents and radicals are as follows:

The power rule states that for any non-negative integer $n$ , $(ab)^n = a^nb^n$ .
The property $\sqrt[n]{a^n} = a$ can be used to simplify radicals.
Absolute value expressions can be simplified by removing the absolute value sign and keeping the expression inside the same.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, follow these steps:

Simplify the fraction inside the radical expression.
Apply the power rule to simplify the variables inside the radical expression.
Simplify the radicals using the property $\sqrt[n]{a^n} = a$ .
Simplify the absolute value expression.

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

A: A rational exponent is an exponent that is a fraction, while a radical expression is an expression that involves a root or a power of a number. For example, $a^{\frac{1}{2}}$ is a rational exponent, while $\sqrt{a}$ is a radical expression.

Q: Can I simplify a radical expression with a negative exponent?

A: Yes, you can simplify a radical expression with a negative exponent. To do this, follow these steps:

Rewrite the negative exponent as a positive exponent by changing the sign of the exponent.
Simplify the radical expression using the properties of exponents and radicals.

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

A: To simplify a radical expression with a variable in the radicand, follow these steps:

Factor the radicand to find the greatest perfect square factor.
Rewrite the radicand as the product of the greatest perfect square factor and the remaining factor.
Simplify the radical expression using the properties of exponents and radicals.

Q: Can I simplify a radical expression with a variable in the index?

A: Yes, you can simplify a radical expression with a variable in the index. To do this, follow these steps:

Rewrite the index as a rational exponent.
Simplify the radical expression using the properties of exponents and radicals.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill in mathematics. By understanding the properties of exponents and radicals, you can simplify complex expressions and arrive at the final answer. We hope this Q&A guide has helped you better understand the concept of simplifying radical expressions.

Practice Problems

Simplify the radical expression: $\sqrt[3]{\frac{64}{27} a^9 b^{18} c^{27}}$ .
Simplify the radical expression: $\sqrt[5]{\frac{100}{32} a^{15} b^{30} c^{40}}$ .
Simplify the radical expression: $\sqrt[4]{\frac{81}{16} a^8 b^{12} c^{16}}$ .

References

[1] "Algebra and Trigonometry" by Michael Sullivan
[2] "Mathematics for the Nonmathematician" by Morris Kline
[3] "Calculus" by Michael Spivak

Discussion

Key Takeaways

Simplifying radical expressions involves applying the properties of exponents and radicals.
The power rule states that for any non-negative integer $n$ , $(ab)^n = a^nb^n$ .
The property $\sqrt[n]{a^n} = a$ can be used to simplify radicals.
Absolute value expressions can be simplified by removing the absolute value sign and keeping the expression inside the same.