Assessment and Learning in Knowledge Spaces (ALEKS) Basic Math Placement Practice Test

What is the result when you add two numbers that have the same exponent?

1. You can directly add them

2. You must factor them first

3. They must be simplified to a common base

4. You cannot add them together

The correct answer is: You cannot add them together

When adding two numbers with the same exponent, it’s important to understand that the operation cannot be performed simply by directly adding the base numbers together. This stems from the fundamental properties of exponents and how they work mathematically. When two numbers (such as \(a^n\) and \(b^n\)) share the same exponent \(n\), they cannot be combined through direct addition unless their bases are the same. If the bases are different, such as \(2^3 + 3^3\), you cannot just add the resulting values \(8 + 27\) in terms of their exponential forms without first calculating their individual values. Thus, they cannot simply be added as one might do with regular numbers, leading to the conclusion that this type of addition is not valid. Understanding this distinction is crucial for working with algebraic expressions involving exponents, as it reinforces the need to handle such scenarios with care and clarity, ensuring each unique value is dealt with on its own before any combination is attempted.