Rewrite Each Item To Expressions With Positive Exponents

Rewriting Expressions with Negative Exponents: A Comprehensive Guide

Negative exponents can seem daunting at first, but mastering them is crucial for success in algebra and beyond. This comprehensive guide will walk you through the process of rewriting expressions containing negative exponents into expressions with only positive exponents. We'll cover various scenarios, from simple terms to complex expressions involving multiple variables and fractions. By the end, you'll be confident in tackling any negative exponent problem.

Understanding Negative Exponents

The fundamental rule governing negative exponents is this: a^-n = 1/aⁿ, where 'a' is any non-zero number and 'n' is any positive integer. This rule essentially states that a negative exponent indicates a reciprocal. To eliminate the negative exponent, you move the base to the denominator (if it's in the numerator) or to the numerator (if it's in the denominator) and change the sign of the exponent to positive.

Example 1: Simple Term

Let's start with a simple example: x^-3. Applying the rule, we get 1/x³. The negative exponent is gone, and we now have an equivalent expression with a positive exponent.

Example 2: Numerical Base

Consider the expression 5^-2. Using the rule, this becomes 1/5², which simplifies to 1/25.

Rewriting Expressions with Multiple Terms

Things get slightly more complex when dealing with expressions containing multiple terms. The key is to apply the rule to each term individually, ensuring that you carefully handle any coefficients and other variables.

Example 3: Multiple Terms in the Numerator

Let's examine the expression 3x^-2y⁴. To rewrite this with positive exponents, we apply the rule to the term with the negative exponent: 3y⁴/x². The x^-2 term moved to the denominator, becoming x², while the rest of the expression remained in the numerator.

Example 4: Multiple Terms in the Denominator

Consider this expression: 4/(x^-1y²z^-3). This expression has negative exponents in the denominator. To get rid of them, we move each term with a negative exponent to the numerator, changing the sign of its exponent. This gives us 4x¹z³/y², which simplifies to 4xz³/y².

Rewriting Expressions with Fractions

When dealing with fractions containing negative exponents, a strategic approach is necessary. Often, it's helpful to first simplify the fraction before applying the rule for negative exponents.

Example 5: Fraction with Negative Exponent in the Numerator

Let's analyze the fraction (x^-2y³)/(z⁴). We can rewrite this as: (y³)/(x²z⁴). The x^-2 moved to the denominator, changing its exponent to positive.

Example 6: Fraction with Negative Exponents in the Numerator and Denominator

Consider the expression (2x^-3y²)/(3z^-1w⁴). We can rewrite this as: (2y²z¹)/(3x³w⁴), which simplifies to (2y²z)/(3x³w⁴). Both negative exponents were addressed by moving the terms accordingly.

Rewriting Expressions with Parentheses and Powers

Parentheses and powers add another layer of complexity. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction. Always address the exponents within parentheses first before dealing with the exponents outside the parentheses.

Example 7: Parentheses and Powers

Let's tackle (2x^-1y²)^-3. First, we address the exponent outside the parenthesis by applying it to each term inside the parenthesis: 2^-3x^(-1)(-3)y^(2)(-3). This simplifies to 2^-3x³y^-6. Finally, we deal with the negative exponents, moving 2^-3 to the denominator and y^-6 to the denominator: x³/(8y⁶).

Example 8: Nested Parentheses

A more challenging example: [(x^-2y³)²]^-1. We work from the inside out. First, we apply the inner exponent: (x^-4y⁶)^-1. Next, we apply the outer exponent: x⁴y^-6. Finally, we deal with the negative exponent: x⁴/y⁶.

Combining Techniques: A Comprehensive Example

Let's conclude with a more complex example that integrates all the techniques discussed:

(3x^-2y⁴z^-1)/(2a³b^-2c^-4)^-1

1. Inner exponent: First, address the exponent -1 applied to the denominator. This reverses the sign of all exponents within the parenthesis:

   (3x^-2y⁴z^-1)/(2^-1a^-3b²c⁴)

2. Negative exponents in the denominator: Move terms with negative exponents from the denominator to the numerator:

   (3x^-2y⁴z^-1 * 2¹a³b^-2c⁴)

3. Simplify:

   (6x^-2y⁴z^-1a³b^-2c⁴)

4. Move terms with negative exponents to the denominator:

   (6a³c⁴y⁴)/(x²z¹b²)

Practice Makes Perfect

Rewriting expressions with negative exponents is a skill that improves with practice. The key is to understand the fundamental rule and apply it systematically. Start with simple examples and gradually work your way up to more complex expressions. Remember to always check your work and ensure your final answer contains only positive exponents. Through consistent practice, you'll master this crucial algebraic skill.