Which Expression Is Equivalent To ${ \left(\frac{\left(3 X Y {-5}\right) 3}{\left(x^{-2} Y 2\right) {-4}}\right)^{-2} }$ Assume { X \neq 0, Y \neq 0 $}$.A. { \frac{x^{10} Y^{14}}{729}$} B . \[ B. \[ B . \[ \frac{x^{22}}{18

Introduction

Algebraic expressions can be complex and daunting, especially when they involve multiple variables and exponents. However, with a clear understanding of the rules of exponents and a systematic approach, we can simplify even the most complex expressions. In this article, we will explore how to simplify the expression $\left(\frac{\left(3 x y^{-5}\right)^3}{\left(x^{-2} y^2\right)^{-4}}\right)^{-2}$ and determine which of the given options is equivalent to it.

Understanding the Rules of Exponents

Before we dive into the simplification process, let's review the rules of exponents that we will need to apply:

• Product of Powers Rule: When multiplying two powers with the same base, we add the exponents. For example, $x^a \cdot x^b = x^{a+b}$.
• Power of a Power Rule: When raising a power to another power, we multiply the exponents. For example, $(x^a)^b = x^{ab}$.
• Quotient of Powers Rule: When dividing two powers with the same base, we subtract the exponents. For example, $\frac{x^a}{x^b} = x^{a-b}$.
• Negative Exponent Rule: A negative exponent indicates that we need to take the reciprocal of the base. For example, $x^{-a} = \frac{1}{x^a}$.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's apply them to simplify the given expression.

Step 1: Apply the Power of a Power Rule

We start by applying the power of a power rule to the numerator and denominator of the expression:

$$\left(\frac{\left(3 x y^{-5}\right)^3}{\left(x^{-2} y^2\right)^{-4}}\right)^{-2} = \frac{\left(3 x y^{-5}\right)^3}{\left(x^{-2} y^2\right)^{-4}}^{-2}$$

Using the power of a power rule, we can rewrite the expression as:

$$\frac{\left(3 x y^{-5}\right)^3}{\left(x^{-2} y^2\right)^{-4}}^{-2} = \frac{3^3 x^3 y^{-15}}{x^8 y^{-8}}^{-2}$$

Step 2: Apply the Negative Exponent Rule

Next, we apply the negative exponent rule to simplify the expression further:

$$\frac{3^3 x^3 y^{-15}}{x^8 y^{-8}}^{-2} = \frac{3^3 x^3 y^{15}}{x^8 y^8}^{-2}$$

Using the negative exponent rule, we can rewrite the expression as:

$$\frac{3^3 x^3 y^{15}}{x^8 y^8}^{-2} = \frac{27 x^3 y^{15}}{x^8 y^8}^{-2}$$

Step 3: Apply the Quotient of Powers Rule

Now, we apply the quotient of powers rule to simplify the expression further:

$$\frac{27 x^3 y^{15}}{x^8 y^8}^{-2} = \frac{27 x^{3-8} y^{15-8}}{1}^{-2}$$

Using the quotient of powers rule, we can rewrite the expression as:

$$\frac{27 x^{3-8} y^{15-8}}{1}^{-2} = \frac{27 x^{-5} y^7}{1}^{-2}$$

Step 4: Apply the Negative Exponent Rule Again

Finally, we apply the negative exponent rule again to simplify the expression further:

$$\frac{27 x^{-5} y^7}{1}^{-2} = 27 x^5 y^{-7}$$

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options to determine which one is equivalent to it.

Option A

Option A is $\frac{x^{10} y^{14}}{729}$. To determine if this option is equivalent to the simplified expression, we need to multiply the numerator and denominator by $729$:

$$\frac{x^{10} y^{14}}{729} = \frac{729 x^{10} y^{14}}{729^2}$$

Using the product of powers rule, we can rewrite the expression as:

$$\frac{729 x^{10} y^{14}}{729^2} = \frac{729 x^{10} y^{14}}{531441}$$

This expression is not equivalent to the simplified expression.

Option B

Option B is $\frac{x^{22}}{18}$. To determine if this option is equivalent to the simplified expression, we need to multiply the numerator and denominator by $18$:

$$\frac{x^{22}}{18} = \frac{18 x^{22}}{18^2}$$

Using the product of powers rule, we can rewrite the expression as:

$$\frac{18 x^{22}}{18^2} = \frac{18 x^{22}}{324}$$

This expression is not equivalent to the simplified expression.

Conclusion

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Q: What are the rules of exponents that I need to know to simplify complex algebraic expressions?

A: The rules of exponents that you need to know to simplify complex algebraic expressions are:

• Product of Powers Rule: When multiplying two powers with the same base, we add the exponents. For example, $x^a \cdot x^b = x^{a+b}$.
• Power of a Power Rule: When raising a power to another power, we multiply the exponents. For example, $(x^a)^b = x^{ab}$.
• Quotient of Powers Rule: When dividing two powers with the same base, we subtract the exponents. For example, $\frac{x^a}{x^b} = x^{a-b}$.
• Negative Exponent Rule: A negative exponent indicates that we need to take the reciprocal of the base. For example, $x^{-a} = \frac{1}{x^a}$.

Q: How do I apply the power of a power rule to simplify an expression?

A: To apply the power of a power rule, you need to multiply the exponents when raising a power to another power. For example, $(x^a)^b = x^{ab}$.

Q: How do I apply the negative exponent rule to simplify an expression?

A: To apply the negative exponent rule, you need to take the reciprocal of the base when you see a negative exponent. For example, $x^{-a} = \frac{1}{x^a}$.

Q: How do I apply the quotient of powers rule to simplify an expression?

A: To apply the quotient of powers rule, you need to subtract the exponents when dividing two powers with the same base. For example, $\frac{x^a}{x^b} = x^{a-b}$.

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

A: A positive exponent indicates that we need to multiply the base by itself as many times as the exponent says. For example, $x^a = x \cdot x \cdot x \cdot ... \cdot x$ (a times). A negative exponent indicates that we need to take the reciprocal of the base. For example, $x^{-a} = \frac{1}{x^a}$.

Q: How do I simplify an expression with multiple variables and exponents?

A: To simplify an expression with multiple variables and exponents, you need to apply the rules of exponents in the correct order. First, apply the power of a power rule, then the negative exponent rule, and finally the quotient of powers rule.

Q: What is the final answer to the expression $\left(\frac{\left(3 x y^{-5}\right)^3}{\left(x^{-2} y^2\right)^{-4}}\right)^{-2}$?

A: The final answer to the expression $\left(\frac{\left(3 x y^{-5}\right)^3}{\left(x^{-2} y^2\right)^{-4}}\right)^{-2}$ is $27 x^5 y^{-7}$.

Q: How do I know which option is equivalent to the simplified expression?

A: To determine which option is equivalent to the simplified expression, you need to multiply the numerator and denominator of each option by the same value as the simplified expression. If the resulting expression is the same as the simplified expression, then the option is equivalent.

Q: What if I get stuck on a problem and don't know how to simplify it?

A: If you get stuck on a problem and don't know how to simplify it, try breaking it down into smaller parts and applying the rules of exponents one at a time. You can also try using a different approach or looking for a pattern in the expression. If you're still stuck, you can ask for help from a teacher or tutor.