Given X \textgreater 0 X\ \textgreater \ 0 X \textgreater 0 And Y \textgreater 0 Y\ \textgreater \ 0 Y \textgreater 0 , Select The Expression That Is Equivalent To 81 X 8 Y 12 4 \sqrt[4]{81 X^8 Y^{12}} 4 81 X 8 Y 12 ​ .A. 9 X 2 Y 3 9 X^2 Y^3 9 X 2 Y 3 B. 3 X 2 Y 3 3 X^2 Y^3 3 X 2 Y 3 C. 3 X 4 Y 8 3 X^4 Y^8 3 X 4 Y 8 D. 9 X 4 Y 8 9 X^4 Y^8 9 X 4 Y 8

Understanding the Problem

Given the expression $\sqrt[4]{81 x^8 y^{12}}$, we are asked to select the equivalent expression from the given options. To simplify this expression, we need to apply the properties of radicals and exponents.

Breaking Down the Expression

The given expression can be broken down into two parts: the radicand and the index. The radicand is $81 x^8 y^{12}$, and the index is $4$. To simplify the expression, we need to apply the property of radicals that states $\sqrt[n]{a^n} = a$.

Simplifying the Radicand

The radicand can be simplified by expressing $81$ as a power of $3$. We know that $81 = 3^4$, so we can rewrite the radicand as:

$$81 x^8 y^{12} = (3^4) x^8 y^{12}$$

Applying the Property of Radicals

Now, we can apply the property of radicals that states $\sqrt[n]{a^n} = a$. In this case, we have:

$$\sqrt[4]{(3^4) x^8 y^{12}} = 3^1 x^2 y^3$$

Simplifying the Expression

The expression $3^1 x^2 y^3$ can be simplified further by applying the property of exponents that states $a^m \cdot a^n = a^{m+n}$. In this case, we have:

$$3^1 x^2 y^3 = 3 x^2 y^3$$

Comparing with the Options

Now, we can compare the simplified expression with the given options:

• A. $9 x^2 y^3$
• B. $3 x^2 y^3$
• C. $3 x^4 y^8$
• D. $9 x^4 y^8$

The only option that matches the simplified expression is:

• B. $3 x^2 y^3$

Conclusion

In this article, we have simplified the expression $\sqrt[4]{81 x^8 y^{12}}$ by applying the properties of radicals and exponents. We have shown that the simplified expression is equivalent to $3 x^2 y^3$. This result is consistent with the given options, and we can conclude that the correct answer is option B.

Key Takeaways

• The property of radicals states that $\sqrt[n]{a^n} = a$.
• The property of exponents states that $a^m \cdot a^n = a^{m+n}$.
• To simplify a radical expression, we need to apply the properties of radicals and exponents.

Practice Problems

1. Simplify the expression $\sqrt[3]{64 x^9 y^{18}}$.
2. Simplify the expression $\sqrt[2]{16 x^4 y^8}$.
3. Simplify the expression $\sqrt[5]{32 x^{10} y^{20}}$.

Answers

1. $4 x^3 y^6$
2. $4 x^2 y^4$
3. $2 x^2 y^4$

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline
  Simplifying Radical Expressions: A Q&A Guide =====================================================

Q: What is the property of radicals that states $\sqrt[n]{a^n} = a$?

A: This property states that when we have a radical expression with an index of $n$ and a radicand that is a power of $a$, we can simplify the expression by taking the $n$th root of the radicand.

Q: How do we apply the property of radicals to simplify an expression?

A: To apply the property of radicals, we need to identify the radicand and the index of the radical expression. We then need to express the radicand as a power of a number, and take the $n$th root of that number.

Q: What is the property of exponents that states $a^m \cdot a^n = a^{m+n}$?

A: This property states that when we multiply two numbers with the same base, we can add their exponents.

Q: How do we apply the property of exponents to simplify an expression?

A: To apply the property of exponents, we need to identify the base and the exponents of the numbers in the expression. We then need to add the exponents of the numbers with the same base.

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

A: A radical expression is an expression that contains a root, such as $\sqrt[n]{a}$. An exponential expression is an expression that contains a power, such as $a^m$.

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

A: To simplify a radical expression with a negative exponent, we need to express the negative exponent as a positive exponent by taking the reciprocal of the base.

Q: What is the rule for multiplying radical expressions?

A: The rule for multiplying radical expressions is that we can multiply the radicands and keep the same index.

Q: What is the rule for dividing radical expressions?

A: The rule for dividing radical expressions is that we can divide the radicands and keep the same index.

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

A: To simplify a radical expression with a variable in the radicand, we need to express the variable as a power of a number, and then apply the property of radicals.

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

A: A rational exponent is an exponent that can be expressed as a fraction, such as $a^{1/2}$. An irrational exponent is an exponent that cannot be expressed as a fraction, such as $a^{\sqrt{2}}$.

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

A: To simplify a radical expression with a rational exponent, we need to express the rational exponent as a power of a number, and then apply the property of radicals.

Q: What is the rule for simplifying a radical expression with a rational exponent?

A: The rule for simplifying a radical expression with a rational exponent is that we can express the rational exponent as a power of a number, and then apply the property of radicals.

Q: How do we simplify a radical expression with a variable in the exponent?

A: To simplify a radical expression with a variable in the exponent, we need to express the variable as a power of a number, and then apply the property of radicals.

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

A: A radical expression is an expression that contains a root, such as $\sqrt[n]{a}$. An algebraic expression is an expression that contains variables and constants, such as $ax^2 + bx + c$.

Q: How do we simplify a radical expression with a variable in the algebraic expression?

A: To simplify a radical expression with a variable in the algebraic expression, we need to express the variable as a power of a number, and then apply the property of radicals.

Q: What is the rule for simplifying a radical expression with a variable in the algebraic expression?

A: The rule for simplifying a radical expression with a variable in the algebraic expression is that we can express the variable as a power of a number, and then apply the property of radicals.

Conclusion

In this article, we have answered some common questions about simplifying radical expressions. We have covered topics such as the property of radicals, the property of exponents, and the rules for multiplying and dividing radical expressions. We have also covered topics such as rational exponents, irrational exponents, and algebraic expressions. By following these rules and guidelines, you can simplify radical expressions with ease.

Practice Problems

1. Simplify the expression $\sqrt[3]{64 x^9 y^{18}}$.
2. Simplify the expression $\sqrt[2]{16 x^4 y^8}$.
3. Simplify the expression $\sqrt[5]{32 x^{10} y^{20}}$.

Answers

1. $4 x^3 y^6$
2. $4 x^2 y^4$
3. $2 x^2 y^4$

References

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