How to Add Numbers with the Same Base and Different Exponents?

To add numbers with the same base but different exponents, you generally need to manipulate the expressions so they have the same exponent. This is analogous to finding a common denominator when adding fractions. Here's a breakdown:

Steps to Add Numbers with the Same Base and Different Exponents

1. Adjust the Exponents: Modify the numbers so they all have the same exponent. This often involves rewriting the numbers using exponent rules. For example, consider the expression (base^exponent1) + (base^exponent2). You might need to rewrite one of the terms so that both terms share the same exponent.

2. Factor out common term: Factoring out a common base and exponent helps to organize your calculation

3. Add the Coefficients (Mantissas): After adjusting the exponents, you can essentially factor out the common base^exponent. Then, add the remaining coefficients (mantissas).

4. Simplify: After summing the coefficients, perform the arithmetic operations in your math expression.

Example:

Let's say you want to add 2³ + 2⁵.

1. Adjust the Exponents:

   • We can express 2⁵ as 2² 2³ = 4 2³.

2. Add the Coefficients:

   • Now we have 2³ + 4 * 2³.
   • Factor out 2³: (1 + 4) 2³ = 5 2³.

3. Simplify:

   • 5 2³ = 5 8 = 40.

Therefore, 2³ + 2⁵ = 40.

Analogy to Adding Fractions

The reference highlights the similarity to adding fractions. When adding fractions, you need a common denominator. Similarly, when adding numbers with the same base and different exponents, you need a common exponent. Then you perform the addition on the numerators or mantissas (coefficients of the terms with the same exponent) respectively.