Problem 35 - Algebra 1

Question

Rewrite the expression with positive exponents. $$8 x^{-2} y^{-6}$$

Step-by-Step Solution

Answer

The given expression \(8x^{-2}y^{-6}\) with positive exponents is \(8/(x^2 * y^6)\).

Step 1: Identify Negative Exponents

In the given expression \(8x^{-2}y^{-6}\), identify the terms with negative exponents. They are \(x^{-2}\) and \(y^{-6}\).

Step 2: Rewrite these terms with positive exponents

For \(x^{-2}\), write this term as \(1/x^2\). And for \(y^{-6}\), write it as \(1/y^6\). So, our expression becomes \(8 * (1/x^2) * (1/y^6)\).

Step 3: Simplify the Expression

Once we have converted the terms with negative exponents into positive, further simplification results in the final expression: \(8/(x^2 * y^6)\).

Key Concepts

Understanding Negative Exponents

Negative exponents refer to the situation where a base is raised to a negative power. When you see a term like \( x^{-n} \), it means that you're dealing with the reciprocal of the base raised to the positive exponent. In simpler terms:

• \( x^{-n} = \frac{1}{x^n} \)

This rule is very important because it allows for the transformation of expressions in a way that simplifies multiplication and division.
Whenever you encounter a negative exponent, remember that your main job is to flip the base to the other side of the fraction (top to bottom or bottom to top) and then make the exponent positive.
For example, in the expression \( 8x^{-2}y^{-6} \), the terms with negative exponents, \( x^{-2} \) and \( y^{-6} \), are rewritten as \( \frac{1}{x^2} \) and \( \frac{1}{y^6} \), respectively. This transforms the initial expression into a more manageable form, free of negative exponents.

Expression Simplification

Expression simplification involves the reduction or transformation of a mathematical expression into its simplest form. This process is crucial when dealing with more complex algebraic or polynomial expressions.
For simplification, always start by identifying components that can be combined, canceled, or restructured. In the case of an expression containing terms with negative exponents, simplifying means first making all exponents positive.
In our exercise, after transforming the negative exponents to \( \frac{1}{x^2} \) and \( \frac{1}{y^6} \), we simplify by multiplying:

• The numerical coefficient \( 8 \)
• With the newly positive exponent terms \( \frac{1}{x^2} \) and \( \frac{1}{y^6} \)

This results in the expression \( \frac{8}{x^2y^6} \), which is fully simplified and contains no negative exponents. Just one tip: always take care of parentheses and maintain the order of operations (PEMDAS/BODMAS) to ensure error-free results.

Working With Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. They are the basic building blocks in algebra and often require rewriting and simplification, especially when negative exponents are involved.
An algebraic expression like \( 8x^{-2}y^{-6} \) represents a continuous multiplication of its components. The rule of handling exponents greatly influences our ability to manipulate these expressions:

• Identify: Recognize terms with exponents, whether negative or positive.
• Rewrite: Convert terms with negative exponents using the reciprocal and make exponents positive.
• Simplify: Combine like terms, reduce the expression, or carry out arithmetic operations.

Don't be intimidated by complex-looking algebraic expressions. Understand each part and handle them step by step. Keep working through examples and practicing these transformations, and you'll find that handling algebraic expressions with ease becomes second nature.