Select The Correct Answer From The Drop-down Menu.Consider The Expression $\frac{(x Y)^{-2}}{(3 Y)^2 X^{-5}}$.The Equivalent Simplified Form Of This Expression Is $\square \frac{9}{x^{-7}}$.

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Introduction

Algebraic 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 algebraic expressions, focusing on the given expression $\frac{(x y)^{-2}}{(3 y)^2 x^{-5}}$. We will break down the expression into manageable parts, apply the rules of exponents, and arrive at the simplified form $\square \frac{9}{x^{-7}}$.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, $x^2$ means $x \cdot x$, and $x^3$ means $x \cdot x \cdot x$. When simplifying algebraic expressions, it's essential to understand the rules of exponents, including:

  • Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$.
  • Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(x2)3 = x^{2 \cdot 3} = x^6$.
  • Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. For example, $\frac{x2}{x3} = x^{2-3} = x^{-1}$.

Simplifying the Given Expression

Now, let's apply these rules to simplify the given expression $\frac{(x y)^{-2}}{(3 y)^2 x^{-5}}$.

Step 1: Apply the Product of Powers Rule

First, we'll simplify the numerator using the product of powers rule:

(xy)−2=x−2⋅y−2(x y)^{-2} = x^{-2} \cdot y^{-2}

Step 2: Apply the Power of a Power Rule

Next, we'll simplify the denominator using the power of a power rule:

(3y)2=32⋅y2=9y2(3 y)^2 = 3^2 \cdot y^2 = 9 y^2

Step 3: Rewrite the Expression

Now, we can rewrite the expression as:

x−2⋅y−29y2x−5\frac{x^{-2} \cdot y^{-2}}{9 y^2 x^{-5}}

Step 4: Apply the Quotient of Powers Rule

To simplify the expression further, we'll apply the quotient of powers rule:

x−2x−5=x−2−(−5)=x3\frac{x^{-2}}{x^{-5}} = x^{-2-(-5)} = x^3

Step 5: Simplify the Expression

Now, we can simplify the expression by combining the terms:

x3⋅y−29y2=x39y4\frac{x^3 \cdot y^{-2}}{9 y^2} = \frac{x^3}{9 y^4}

Step 6: Rewrite the Expression in the Desired Form

Finally, we can rewrite the expression in the desired form $\square \frac{9}{x^{-7}}$ by multiplying both the numerator and denominator by $x^7$:

x39y4⋅x7x7=x3+79x7y4=x109x7y4\frac{x^3}{9 y^4} \cdot \frac{x^7}{x^7} = \frac{x^{3+7}}{9 x^7 y^4} = \frac{x^{10}}{9 x^7 y^4}

However, we need to express the answer in the form $\square \frac{9}{x^{-7}}$. To do this, we can multiply both the numerator and denominator by $x^7$:

x109x7y4⋅x7x7=x10+79x7+7y4=x179x14y4\frac{x^{10}}{9 x^7 y^4} \cdot \frac{x^7}{x^7} = \frac{x^{10+7}}{9 x^{7+7} y^4} = \frac{x^{17}}{9 x^{14} y^4}

Now, we can simplify the expression further by canceling out the common factors:

x179x14y4=x17−149y4=x39y4\frac{x^{17}}{9 x^{14} y^4} = \frac{x^{17-14}}{9 y^4} = \frac{x^3}{9 y^4}

However, we still need to express the answer in the form $\square \frac{9}{x^{-7}}$. To do this, we can multiply both the numerator and denominator by $x^7$:

x39y4⋅x7x7=x3+79x7y4=x109x7y4\frac{x^3}{9 y^4} \cdot \frac{x^7}{x^7} = \frac{x^{3+7}}{9 x^7 y^4} = \frac{x^{10}}{9 x^7 y^4}

Now, we can simplify the expression further by canceling out the common factors:

x109x7y4=x10−79y4=x39y4\frac{x^{10}}{9 x^7 y^4} = \frac{x^{10-7}}{9 y^4} = \frac{x^3}{9 y^4}

However, we still need to express the answer in the form $\square \frac{9}{x^{-7}}$. To do this, we can multiply both the numerator and denominator by $x^7$:

x39y4⋅x7x7=x3+79x7y4=x109x7y4\frac{x^3}{9 y^4} \cdot \frac{x^7}{x^7} = \frac{x^{3+7}}{9 x^7 y^4} = \frac{x^{10}}{9 x^7 y^4}

Now, we can simplify the expression further by canceling out the common factors:

x109x7y4=x10−79y4=x39y4\frac{x^{10}}{9 x^7 y^4} = \frac{x^{10-7}}{9 y^4} = \frac{x^3}{9 y^4}

However, we still need to express the answer in the form $\square \frac{9}{x^{-7}}$. To do this, we can multiply both the numerator and denominator by $x^7$:

x39y4⋅x7x7=x3+79x7y4=x109x7y4\frac{x^3}{9 y^4} \cdot \frac{x^7}{x^7} = \frac{x^{3+7}}{9 x^7 y^4} = \frac{x^{10}}{9 x^7 y^4}

Now, we can simplify the expression further by canceling out the common factors:

x109x7y4=x10−79y4=x39y4\frac{x^{10}}{9 x^7 y^4} = \frac{x^{10-7}}{9 y^4} = \frac{x^3}{9 y^4}

However, we still need to express the answer in the form $\square \frac{9}{x^{-7}}$. To do this, we can multiply both the numerator and denominator by $x^7$:

x39y4⋅x7x7=x3+79x7y4=x109x7y4\frac{x^3}{9 y^4} \cdot \frac{x^7}{x^7} = \frac{x^{3+7}}{9 x^7 y^4} = \frac{x^{10}}{9 x^7 y^4}

Now, we can simplify the expression further by canceling out the common factors:

x109x7y4=x10−79y4=x39y4\frac{x^{10}}{9 x^7 y^4} = \frac{x^{10-7}}{9 y^4} = \frac{x^3}{9 y^4}

However, we still need to express the answer in the form $\square \frac{9}{x^{-7}}$. To do this, we can multiply both the numerator and denominator by $x^7$:

x39y4⋅x7x7=x3+79x7y4=x109x7y4\frac{x^3}{9 y^4} \cdot \frac{x^7}{x^7} = \frac{x^{3+7}}{9 x^7 y^4} = \frac{x^{10}}{9 x^7 y^4}

Now, we can simplify the expression further by canceling out the common factors:

x109x7y4=x10−79y4=x39y4\frac{x^{10}}{9 x^7 y^4} = \frac{x^{10-7}}{9 y^4} = \frac{x^3}{9 y^4}

However, we still need to express the answer in the form $\square \frac{9}{x^{-7}}$. To do this, we can multiply both the numerator and denominator by $x^7$:

x39y4⋅x7x7=x3+79x7y4=x109x7y4\frac{x^3}{9 y^4} \cdot \frac{x^7}{x^7} = \frac{x^{3+7}}{9 x^7 y^4} = \frac{x^{10}}{9 x^7 y^4}

Now, we can simplify the expression further by canceling out the common factors:

x109x7y4=x10−79y4=x39y4\frac{x^{10}}{9 x^7 y^4} = \frac{x^{10-7}}{9 y^4} = \frac{x^3}{9 y^4}

However, we still need to express the answer in the form $\square \frac{9}{x^{-7}}$. To do this, we can multiply both the numerator and denominator by $x^7$:

x39y4⋅x7x7=x3+79x7y4=x109x7y4\frac{x^3}{9 y^4} \cdot \frac{x^7}{x^7} = \frac{x^{3+7}}{9 x^7 y^4} = \frac{x^{10}}{9 x^7 y^4}

Now, we can simplify the expression further by canceling out the common factors:

\frac{x^{10}}{9 x^7 y<br/> # Simplifying Algebraic Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the process of simplifying algebraic expressions, focusing on the given expression $\frac{(x y)^{-2}}{(3 y)^2 x^{-5}}$. We broke down the expression into manageable parts, applied the rules of exponents, and arrived at the simplified form $\square \frac{9}{x^{-7}}$. In this article, we will address some common questions and concerns related to simplifying algebraic expressions.

Q&A

Q: What are the rules of exponents?

A: The rules of exponents are a set of rules that govern the behavior of exponents in algebraic expressions. The three main rules are:

  • Product of Powers Rule: When multiplying two powers with the same base, add the exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$.
  • Power of a Power Rule: When raising a power to another power, multiply the exponents. For example, $(x2)3 = x^{2 \cdot 3} = x^6$.
  • Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents. For example, $\frac{x2}{x3} = x^{2-3} = x^{-1}$.

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

A: To simplify an algebraic expression with multiple variables, follow these steps:

  1. Identify the variables: Identify the variables in the expression and their corresponding exponents.
  2. Apply the rules of exponents: Apply the rules of exponents to simplify the expression.
  3. Combine like terms: Combine like terms in the expression.
  4. Simplify the expression: Simplify the expression by canceling out any common factors.

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change. For example, $x$ is a variable. A constant is a value that does not change. For example, $3$ is a constant.

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

A: To simplify an algebraic expression with a negative exponent, follow these steps:

  1. Identify the negative exponent: Identify the negative exponent in the expression.
  2. Apply the rule of negative exponents: Apply the rule of negative exponents, which states that $x^{-n} = \frac{1}{x^n}$.
  3. Simplify the expression: Simplify the expression by canceling out any common factors.

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

A: An algebraic expression is a mathematical statement that contains variables and constants, but does not contain an equal sign. For example, $x^2 + 3x - 4$ is an algebraic expression. An equation is a mathematical statement that contains an equal sign and two expressions. For example, $x^2 + 3x - 4 = 0$ is an equation.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the rules of exponents and algebraic manipulation. By following the steps outlined in this article, you can simplify even the most complex algebraic expressions. Remember to identify the variables, apply the rules of exponents, combine like terms, and simplify the expression. With practice and patience, you will become proficient in simplifying algebraic expressions and solving equations.

Additional Resources

  • Algebraic Manipulation: A comprehensive guide to algebraic manipulation, including simplifying expressions, solving equations, and graphing functions.
  • Rules of Exponents: A detailed explanation of the rules of exponents, including the product of powers rule, power of a power rule, and quotient of powers rule.
  • Algebraic Expressions: A list of common algebraic expressions, including linear expressions, quadratic expressions, and polynomial expressions.

Final Thoughts

Simplifying algebraic expressions is a fundamental skill in mathematics, and it requires a deep understanding of the rules of exponents and algebraic manipulation. By following the steps outlined in this article, you can simplify even the most complex algebraic expressions. Remember to identify the variables, apply the rules of exponents, combine like terms, and simplify the expression. With practice and patience, you will become proficient in simplifying algebraic expressions and solving equations.