Which Expression Is Equivalent To \left(\frac{x^{-4} Y}{x^{-9} Y^5}\right ]? Assume X ≠ 0 , Y ≠ 0 X \neq 0, Y \neq 0 X  = 0 , Y  = 0 .A. Y 8 X 10 \frac{y^8}{x^{10}} X 10 Y 8 ​ B. X 5 Y 7 \frac{x^5}{y^7} Y 7 X 5 ​ C. X 5 Y 4 \frac{x^5}{y^4} Y 4 X 5 ​ D. X Y 7 \frac{x}{y^7} Y 7 X ​

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Introduction

When dealing with algebraic expressions, simplifying exponents is a crucial step in solving equations and inequalities. In this article, we will focus on simplifying the expression (x−4yx−9y5)\left(\frac{x^{-4} y}{x^{-9} y^5}\right), assuming that x≠0x \neq 0 and y≠0y \neq 0. We will explore the properties of exponents and learn how to simplify complex expressions using these properties.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, x3x^3 can be written as x⋅x⋅xx \cdot x \cdot x. When dealing with exponents, it's essential to understand the rules of exponentiation, including the product rule, the quotient rule, and the power rule.

Product Rule

The product rule states that when multiplying two numbers with the same base, we add their exponents. For example, x2⋅x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5.

Quotient Rule

The quotient rule states that when dividing two numbers with the same base, we subtract their exponents. For example, x2x3=x2−3=x−1\frac{x^2}{x^3} = x^{2-3} = x^{-1}.

Power Rule

The power rule states that when raising a power to another power, we multiply their exponents. For example, (x2)3=x2⋅3=x6(x^2)^3 = x^{2 \cdot 3} = x^6.

Simplifying the Expression

Now that we have a solid understanding of the rules of exponentiation, let's apply them to simplify the expression (x−4yx−9y5)\left(\frac{x^{-4} y}{x^{-9} y^5}\right).

Step 1: Simplify the Numerator

The numerator of the expression is x−4yx^{-4} y. We can simplify this by applying the product rule, which states that when multiplying two numbers with the same base, we add their exponents. However, in this case, we have a negative exponent, so we need to apply the rule for negative exponents, which states that x−n=1xnx^{-n} = \frac{1}{x^n}.

x^{-4} y = \frac{1}{x^4} y

Step 2: Simplify the Denominator

The denominator of the expression is x−9y5x^{-9} y^5. We can simplify this by applying the rule for negative exponents, which states that x−n=1xnx^{-n} = \frac{1}{x^n}.

x^{-9} y^5 = \frac{1}{x^9} y^5

Step 3: Simplify the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by dividing the numerator by the denominator.

\frac{x^{-4} y}{x^{-9} y^5} = \frac{\frac{1}{x^4} y}{\frac{1}{x^9} y^5}

To simplify this expression, we can cancel out the common factors in the numerator and denominator.

\frac{\frac{1}{x^4} y}{\frac{1}{x^9} y^5} = \frac{1}{x^4} \cdot \frac{x^9}{y^4} \cdot \frac{y}{y^5}

Now, we can simplify the expression by applying the product rule and the quotient rule.

\frac{1}{x^4} \cdot \frac{x^9}{y^4} \cdot \frac{y}{y^5} = \frac{x^5}{y^7}

Conclusion

In this article, we have learned how to simplify the expression (x−4yx−9y5)\left(\frac{x^{-4} y}{x^{-9} y^5}\right) using the properties of exponents. We have applied the product rule, the quotient rule, and the power rule to simplify the expression and arrive at the final answer, which is x5y7\frac{x^5}{y^7}.

Final Answer

The final answer is x5y7\boxed{\frac{x^5}{y^7}}.

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Introduction

In our previous article, we explored the properties of exponents and learned how to simplify complex expressions using these properties. In this article, we will answer some frequently asked questions about simplifying exponents in algebraic expressions.

Q&A

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

A: A positive exponent represents repeated multiplication, while a negative exponent represents repeated division. For example, x3x^3 represents x⋅x⋅xx \cdot x \cdot x, while x−3x^{-3} represents 1x⋅x⋅x\frac{1}{x \cdot x \cdot x}.

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

A: To simplify an expression with a negative exponent in the numerator, you can rewrite the negative exponent as a positive exponent in the denominator. For example, x−4y3=y3x4\frac{x^{-4}}{y^3} = \frac{y^3}{x^4}.

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

A: To simplify an expression with a negative exponent in the denominator, you can rewrite the negative exponent as a positive exponent in the numerator. For example, x2x−3=x5\frac{x^2}{x^{-3}} = x^5.

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying exponents with the same base, you add their exponents. For example, x2⋅x3=x2+3=x5x^2 \cdot x^3 = x^{2+3} = x^5.

Q: What is the rule for dividing exponents with the same base?

A: When dividing exponents with the same base, you subtract their exponents. For example, x2x3=x2−3=x−1\frac{x^2}{x^3} = x^{2-3} = x^{-1}.

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

A: To simplify an expression with multiple exponents, you can apply the rules of exponentiation in the correct order. For example, (x2)3=x2⋅3=x6(x^2)^3 = x^{2 \cdot 3} = x^6.

Q: What is the rule for raising a power to another power?

A: When raising a power to another power, you multiply their exponents. For example, (x2)3=x2⋅3=x6(x^2)^3 = x^{2 \cdot 3} = x^6.

Examples

Example 1: Simplifying an expression with a negative exponent in the numerator

Simplify the expression x−4y3\frac{x^{-4}}{y^3}.

\frac{x^{-4}}{y^3} = \frac{y^3}{x^4}

Example 2: Simplifying an expression with a negative exponent in the denominator

Simplify the expression x2x−3\frac{x^2}{x^{-3}}.

\frac{x^2}{x^{-3}} = x^5

Example 3: Simplifying an expression with multiple exponents

Simplify the expression (x2)3(x^2)^3.

(x^2)^3 = x^{2 \cdot 3} = x^6

Conclusion

In this article, we have answered some frequently asked questions about simplifying exponents in algebraic expressions. We have covered topics such as positive and negative exponents, multiplying and dividing exponents, and raising a power to another power. By applying the rules of exponentiation, we can simplify complex expressions and arrive at the final answer.

Final Answer

The final answer is x5y7\boxed{\frac{x^5}{y^7}}.