Which Expression Is Equivalent To $\left(9 X^2 Y^6\right)^{-1 / 2}$?A. $3 X Y^3$ B. $\frac{1}{3 X Y^3}$ C. $\frac{3}{x Y^3}$ D. $\frac{x Y^2}{3}$

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Understanding the Problem

When dealing with exponents, it's essential to understand the rules that govern them. In this case, we're given the expression $\left(9 x^2 y^6\right)^{-1 / 2}$ and asked to find an equivalent expression. To solve this problem, we need to apply the rules of exponents, specifically the rule for negative exponents and the rule for fractional exponents.

Applying the Rules of Exponents

The rule for negative exponents states that $a^{-n} = \frac{1}{a^n}$. This means that if we have a negative exponent, we can rewrite it as a fraction with the reciprocal of the base in the numerator and the positive exponent in the denominator.

The rule for fractional exponents states that $a^{m/n} = \sqrt[n]{a^m} = (a^m)^{1/n}$. This means that if we have a fractional exponent, we can rewrite it as a radical with the base raised to the power of the numerator and the denominator as the index of the radical.

Simplifying the Expression

Using the rules of exponents, we can simplify the expression $\left(9 x^2 y^6\right)^{-1 / 2}$ as follows:

$\left(9 x^2 y^6\right)^{-1 / 2} = \frac{1}{\left(9 x^2 y^6\right)^{1/2}}$

Simplifying the Radical

To simplify the radical, we need to apply the rule for fractional exponents. We can rewrite the expression as:

$\frac{1}{\left(9 x^2 y^6\right)^{1/2}} = \frac{1}{\sqrt{9 x^2 y^6}}$

Simplifying the Square Root

To simplify the square root, we need to find the square root of the radicand. We can rewrite the expression as:

$\frac{1}{\sqrt{9 x^2 y^6}} = \frac{1}{\sqrt{9} \cdot \sqrt{x^2} \cdot \sqrt{y^6}}$

Simplifying the Square Roots

To simplify the square roots, we need to apply the rule for square roots. We can rewrite the expression as:

$\frac{1}{\sqrt{9} \cdot \sqrt{x^2} \cdot \sqrt{y^6}} = \frac{1}{3 \cdot x \cdot y^3}$

Simplifying the Fraction

To simplify the fraction, we can rewrite the expression as:

$\frac{1}{3 \cdot x \cdot y^3} = \frac{1}{3 x y^3}$

Conclusion

In conclusion, the expression $\left(9 x^2 y^6\right)^{-1 / 2}$ is equivalent to $\frac{1}{3 x y^3}$.

Answer

The correct answer is B. $\frac{1}{3 x y^3}$.

Discussion

This problem requires a deep understanding of the rules of exponents and how to apply them to simplify expressions. It's essential to remember that negative exponents can be rewritten as fractions with the reciprocal of the base in the numerator and the positive exponent in the denominator. Additionally, fractional exponents can be rewritten as radicals with the base raised to the power of the numerator and the denominator as the index of the radical.

Tips and Tricks

• When dealing with negative exponents, remember that they can be rewritten as fractions with the reciprocal of the base in the numerator and the positive exponent in the denominator.
• When dealing with fractional exponents, remember that they can be rewritten as radicals with the base raised to the power of the numerator and the denominator as the index of the radical.
• When simplifying expressions, always look for opportunities to apply the rules of exponents to simplify the expression.

Related Problems

• Simplify the expression $\left(2 x^3 y^4\right)^{-2 / 3}$.
• Simplify the expression $\left(5 x^2 y^5\right)^{-1 / 4}$.
• Simplify the expression $\left(3 x^4 y^3\right)^{-3 / 2}$.

Solutions

• $\left(2 x^3 y^4\right){-2 / 3} = \frac{1}{\left(2 x^3 y^4\right){2/3}} = \frac{1}{\sqrt[3]{2 x^3 y^4}2} = \frac{1}{\sqrt[3]{4 x^6 y^8}} = \frac{1}{\sqrt[3]{4} \cdot \sqrt[3]{x^6} \cdot \sqrt[3]{y^8}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \sqrt[3]{y^2}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^{8/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4} \cdot x^2 \cdot y^2 \cdot y^{2/3}} = \frac{1}{\sqrt[3]{4
  

Q: What is the rule for negative exponents?

A: The rule for negative exponents states that $a^{-n} = \frac{1}{a^n}$. This means that if we have a negative exponent, we can rewrite it as a fraction with the reciprocal of the base in the numerator and the positive exponent in the denominator.

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

A: To simplify an expression with a negative exponent, we can apply the rule for negative exponents. For example, if we have the expression $\left(2 x^3 y^4\right)^{-2 / 3}$, we can rewrite it as $\frac{1}{\left(2 x^3 y^4\right)^{2/3}}$.

Q: What is the rule for fractional exponents?

A: The rule for fractional exponents states that $a^{m/n} = \sqrt[n]{a^m} = (a^m)^{1/n}$. This means that if we have a fractional exponent, we can rewrite it as a radical with the base raised to the power of the numerator and the denominator as the index of the radical.

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, we can apply the rule for fractional exponents. For example, if we have the expression $\left(5 x^2 y^5\right)^{-1 / 4}$, we can rewrite it as $\frac{1}{\sqrt[4]{5 x^2 y^5}}$.

Q: What is the difference between a negative exponent and a fractional exponent?

A: A negative exponent is a power that is less than 1, while a fractional exponent is a power that is a fraction. For example, $a^{-2}$ is a negative exponent, while $a^{1/2}$ is a fractional exponent.

Q: How do I simplify an expression with both negative and fractional exponents?

A: To simplify an expression with both negative and fractional exponents, we can apply the rules for negative and fractional exponents separately. For example, if we have the expression $\left(3 x^4 y^3\right)^{-3 / 2}$, we can rewrite it as $\frac{1}{\sqrt[2]{3 x^4 y^3}^3}$.

Q: What are some common mistakes to avoid when simplifying expressions with exponents?

A: Some common mistakes to avoid when simplifying expressions with exponents include:

• Forgetting to apply the rule for negative exponents
• Forgetting to apply the rule for fractional exponents
• Not simplifying the expression fully
• Not checking the work for errors

Q: How can I practice simplifying expressions with exponents?

A: You can practice simplifying expressions with exponents by working through examples and exercises in a textbook or online resource. You can also try simplifying expressions with exponents on your own, using a calculator or computer program to check your work.

Q: What are some real-world applications of simplifying expressions with exponents?

A: Simplifying expressions with exponents has many real-world applications, including:

• Calculating the area and volume of shapes
• Determining the rate of change of a function
• Modeling population growth and decay
• Solving problems in physics and engineering

Q: How can I use simplifying expressions with exponents to solve problems in my everyday life?

A: You can use simplifying expressions with exponents to solve problems in your everyday life by applying the rules for negative and fractional exponents to simplify expressions that arise in real-world situations. For example, you can use simplifying expressions with exponents to calculate the area and volume of shapes, determine the rate of change of a function, or model population growth and decay.

Q: What are some common expressions that involve exponents?

A: Some common expressions that involve exponents include:

• $a^2 + b^2$
• $a^3 - b^3$
• $a^4 + b^4$
• $a^5 - b^5$
• $a^6 + b^6$

Q: How can I simplify expressions that involve exponents?

A: You can simplify expressions that involve exponents by applying the rules for negative and fractional exponents. For example, if we have the expression $a^2 + b^2$, we can rewrite it as $(a + b)^2 - 2ab$.

Q: What are some tips for simplifying expressions with exponents?

A: Some tips for simplifying expressions with exponents include:

• Start by simplifying the expression inside the parentheses
• Apply the rule for negative exponents
• Apply the rule for fractional exponents
• Simplify the expression fully
• Check the work for errors

Q: How can I use simplifying expressions with exponents to solve problems in mathematics?

A: You can use simplifying expressions with exponents to solve problems in mathematics by applying the rules for negative and fractional exponents to simplify expressions that arise in mathematical situations. For example, you can use simplifying expressions with exponents to solve problems in algebra, geometry, and calculus.

Q: What are some common mistakes to avoid when simplifying expressions with exponents in mathematics?

A: Some common mistakes to avoid when simplifying expressions with exponents in mathematics include:

• Forgetting to apply the rule for negative exponents
• Forgetting to apply the rule for fractional exponents
• Not simplifying the expression fully
• Not checking the work for errors

Q: How can I practice simplifying expressions with exponents in mathematics?

A: You can practice simplifying expressions with exponents in mathematics by working through examples and exercises in a textbook or online resource. You can also try simplifying expressions with exponents on your own, using a calculator or computer program to check your work.

Q: What are some real-world applications of simplifying expressions with exponents in mathematics?

A: Simplifying expressions with exponents in mathematics has many real-world applications, including:

• Calculating the area and volume of shapes
• Determining the rate of change of a function
• Modeling population growth and decay
• Solving problems in physics and engineering

Q: How can I use simplifying expressions with exponents in mathematics to solve problems in my everyday life?

A: You can use simplifying expressions with exponents in mathematics to solve problems in your everyday life by applying the rules for negative and fractional exponents to simplify expressions that arise in real-world situations. For example, you can use simplifying expressions with exponents in mathematics to calculate the area and volume of shapes, determine the rate of change of a function, or model population growth and decay.