Simplify The Expression And Eliminate Any Negative Exponent(s).(a) 3 X Y − 3 X − 1 Y − 4 \frac{3 X Y^{-3}}{x^{-1} Y^{-4}} X − 1 Y − 4 3 X Y − 3 ​ □ \square □ (b) ( 2 A − 1 B A 3 B − 2 ) − 3 \left(\frac{2 A^{-1} B}{a^3 B^{-2}}\right)^{-3} ( A 3 B − 2 2 A − 1 B ​ ) − 3 □ \square □

Introduction

In algebra, exponents are a crucial concept that helps us simplify complex expressions. However, when we encounter negative exponents, it can be challenging to simplify them. In this article, we will explore how to simplify expressions with negative exponents and eliminate them.

Simplifying Negative Exponents

A negative exponent is a fraction with a negative power. For example, $x^{-3}$ is equivalent to $\frac{1}{x^3}$. To simplify an expression with a negative exponent, we can use the following rules:

• Rule 1: $x^{-n} = \frac{1}{x^n}$
• Rule 2: $\frac{1}{x^{-n}} = x^n$

Let's apply these rules to the given expressions.

(a) $\frac{3 x y^{-3}}{x^{-1} y^{-4}}$

To simplify this expression, we can start by applying the rules for negative exponents.

$\frac{3 x y^{-3}}{x^{-1} y^{-4}} = \frac{3 x}{x^{-1}} \cdot \frac{y^{-3}}{y^{-4}}$

Using Rule 1, we can rewrite the expression as:

$\frac{3 x}{x^{-1}} \cdot \frac{y^{-3}}{y^{-4}} = 3x^2 \cdot y^{3-4}$

Now, we can simplify the exponent by subtracting the exponents:

$y^{3-4} = y^{-1}$

Using Rule 1 again, we can rewrite the expression as:

$3x^2 \cdot y^{-1} = \frac{3x^2}{y}$

Therefore, the simplified expression is $\frac{3x^2}{y}$.

(b) $\left(\frac{2 a^{-1} b}{a^3 b^{-2}}\right)^{-3}$

To simplify this expression, we can start by applying the rules for negative exponents.

$\left(\frac{2 a^{-1} b}{a^3 b^{-2}}\right)^{-3} = \left(\frac{2 b}{a^3 b^{-2} a^{-1}}\right)^{-3}$

Using Rule 1, we can rewrite the expression as:

$\left(\frac{2 b}{a^3 b^{-2} a^{-1}}\right)^{-3} = \left(\frac{2 b}{a^4 b^{-2}}\right)^{-3}$

Now, we can simplify the exponent by applying the rule for negative exponents:

$\left(\frac{2 b}{a^4 b^{-2}}\right)^{-3} = \left(\frac{a^4 b^{-2}}{2 b}\right)^3$

Using Rule 2, we can rewrite the expression as:

$\left(\frac{a^4 b^{-2}}{2 b}\right)^3 = \frac{a^{12} b^{-6}}{8 b^3}$

Now, we can simplify the exponent by subtracting the exponents:

$\frac{a^{12} b^{-6}}{8 b^3} = \frac{a^{12}}{8 b^9}$

Therefore, the simplified expression is $\frac{a^{12}}{8 b^9}$.

Conclusion

In this article, we have explored how to simplify expressions with negative exponents and eliminate them. We have applied the rules for negative exponents to simplify the given expressions and have obtained the simplified expressions. By following these rules, we can simplify complex expressions with negative exponents and make them easier to work with.

Key Takeaways

• A negative exponent is a fraction with a negative power.
• To simplify an expression with a negative exponent, we can use the following rules:
  • $x^{-n} = \frac{1}{x^n}$
  • $\frac{1}{x^{-n}} = x^n$
• To simplify an expression with a negative exponent, we can start by applying the rules for negative exponents and then simplify the exponent by subtracting the exponents.

Practice Problems

1. Simplify the expression $\frac{2 x^{-2} y}{x^3 y^{-1}}$.
2. Simplify the expression $\left(\frac{3 a^{-2} b^2}{a^4 b^{-1}}\right)^{-2}$.

Answer Key

1. $\frac{2 y^2}{x^5}$
2. $\frac{9 a^6 b^3}{81}$
   Frequently Asked Questions (FAQs) about Simplifying Expressions with Negative Exponents =====================================================================================

Q: What is a negative exponent?

A: A negative exponent is a fraction with a negative power. For example, $x^{-3}$ is equivalent to $\frac{1}{x^3}$.

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

A: To simplify an expression with a negative exponent, you can use the following rules:

• $x^{-n} = \frac{1}{x^n}$
• $\frac{1}{x^{-n}} = x^n$

Q: What is the rule for simplifying a fraction with a negative exponent in the numerator?

A: To simplify a fraction with a negative exponent in the numerator, you can rewrite the fraction as:

$\frac{x^{-n}}{y^m} = \frac{y^m}{x^n}$

Q: What is the rule for simplifying a fraction with a negative exponent in the denominator?

A: To simplify a fraction with a negative exponent in the denominator, you can rewrite the fraction as:

$\frac{x^m}{x^{-n}} = x^{m+n}$

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

A: To simplify an expression with multiple negative exponents, you can start by applying the rules for negative exponents and then simplify the exponent by subtracting the exponents.

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

A: A negative exponent is a fraction with a negative power, while a positive exponent is a fraction with a positive power. For example, $x^{-3}$ is equivalent to $\frac{1}{x^3}$, while $x^3$ is equivalent to $x \cdot x \cdot x$.

Q: Can I simplify an expression with a negative exponent by multiplying or dividing?

A: Yes, you can simplify an expression with a negative exponent by multiplying or dividing. For example, $\frac{2 x^{-2} y}{x^3 y^{-1}}$ can be simplified by multiplying or dividing.

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

A: To simplify an expression with a negative exponent and a variable in the denominator, you can start by applying the rules for negative exponents and then simplify the exponent by subtracting the exponents.

Q: What is the rule for simplifying an expression with a negative exponent and a fraction in the denominator?

A: To simplify an expression with a negative exponent and a fraction in the denominator, you can start by applying the rules for negative exponents and then simplify the exponent by subtracting the exponents.

Q: Can I simplify an expression with a negative exponent and a radical in the denominator?

A: Yes, you can simplify an expression with a negative exponent and a radical in the denominator. For example, $\sqrt{x^{-2}}$ can be simplified by applying the rules for negative exponents and radicals.

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

A: To simplify an expression with a negative exponent and a trigonometric function in the denominator, you can start by applying the rules for negative exponents and then simplify the exponent by subtracting the exponents.

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying expressions with negative exponents. We have covered the rules for simplifying expressions with negative exponents, including fractions, variables, and radicals. By following these rules, you can simplify complex expressions with negative exponents and make them easier to work with.

Key Takeaways

• A negative exponent is a fraction with a negative power.
• To simplify an expression with a negative exponent, you can use the following rules:
  • $x^{-n} = \frac{1}{x^n}$
  • $\frac{1}{x^{-n}} = x^n$
• To simplify an expression with multiple negative exponents, you can start by applying the rules for negative exponents and then simplify the exponent by subtracting the exponents.

Practice Problems

1. Simplify the expression $\frac{2 x^{-2} y}{x^3 y^{-1}}$.
2. Simplify the expression $\left(\frac{3 a^{-2} b^2}{a^4 b^{-1}}\right)^{-2}$.
3. Simplify the expression $\sqrt{x^{-2}}$.

Answer Key

1. $\frac{2 y^2}{x^5}$
2. $\frac{9 a^6 b^3}{81}$
3. $\frac{1}{x}$