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

If you have two numbers with the same exponent, how can you combine them?

1. You can add or subtract them

2. You can only multiply them

3. You must divide them

4. You can do nothing with them

The correct answer is: You can add or subtract them

When you have two numbers that share the same exponent, such as \( a^n \) and \( b^n \), you can combine them by adding or subtracting as long as the bases are specified. The core principle behind this is the properties of exponents, which state that when numbers are raised to the same power, they can be combined through addition or subtraction, resulting in a new expression. For instance, if you have \( 3^2 \) and \( 5^2 \), you can combine them into \( 3^2 + 5^2 \), which equals \( 9 + 25 = 34 \). While you cannot simplify \( 3^2 + 5^2 \) to a single exponential term, you can certainly compute their sum. This principle applies across the board, as long as the exponents are the same and the addition or subtraction is clearly defined. Moreover, when one says you can only multiply, divide, or do nothing with numbers that share the same exponent, it overlooks the arithmetic operations of addition and subtraction, which are fundamentally valid with like terms under the same exponent condition. Therefore, combining them through addition or subtraction is indeed the correct approach.