In the diverse realm of algebra, the ability to combine like terms with exponents plays a vital role in simplifying equations and expressions. The feature that distinguishes ‘like terms’ is not only the variable but also the power it’s raised to — jointly referred to as the ‘exponent.’

Despite the seemingly daunting scenario of variables raised to a power, the process of combining like terms with exponents adheres to the same rules that govern combining like terms. This article will provide an in-depth explanation of this process, supplemented with practical examples. For more examples with detailed explanations, visit https://solvelymath.com/articles/combine-like-terms/.

**Defining Like Terms with Exponents**

Like terms are mathematical expressions that have precisely the same variables raised to identical powers. Therefore, in a situation where like terms have exponents, the variable and its exponent must be the same for the terms to be ‘like’ or similar. Like terms can have different numerical coefficients, and it’s these coefficients that will be combined when simplifying the expression.

**Steps for Combining Like Terms with Exponents**

Here is a step-by-step guide on how to merge like terms with exponents:

**1- Identify the Like Terms: **The first step is to identify the like terms in the expression. Remember, like terms must have the same variable and the same exponent.

**2- Add or Subtract the Coefficients:** Next, you will either add or subtract the coefficients of the like terms, depending on the operation sign given before each term.

**3- Write the Combined Term: **Lastly, retain the variable and the exponent and write the combined coefficient. This gives you the combined form of the like terms.

**Examples of Combining Like Terms with Exponents**

**Example 1:**

Suppose we have the expression: 5x² + 7x² – 3x + 2x.

The like terms with exponents are 5x² and 7x², and the like terms without exponents are -3x and 2x.

Combining the coefficients:

- For x² terms, we add: 5x² + 7x² = 12x²
- For the x terms, we add: -3x + 2x = -x

So, the simplified expression is: 12x² – x

**Example 2:**

Consider the expression: 3a³ + 7b² – 4a³ + 5b².

Here, 3a³ and -4a³ are like terms, and 7b² and 5b² are like terms.

By adding or subtracting the coefficients, we get:

3a³ – 4a³ = -a³

7b² + 5b² = 12b²

So, the simplified expression is: -a³ + 12b²

**Tips To Consider When Combining Like Terms With Exponents**

**Cautiously Consider the Signs**

Always take into account the signs (positive/negative) before each term. A change in sign can impact the whole expression, and such changes should be observed carefully to ensure accurate results.

**Practice Regularly**

Like most mathematical processes, becoming proficient at combining like terms with exponents comes with practice. Regularly solving such problems enables you to better recognize and combine like terms, ultimately enhancing your skill in simplifying complex algebraic expressions.

**Conclusion**

Combining like terms with exponents is fundamental to advanced mathematical computation, as it helps bring complexity to simplicity. It is a transferable skill that not only aids in directly solving problems but also indirectly facilitates the solving of more nuanced mathematical problems. Despite the potential initial difficulties, one can master this aspect of algebra by understanding the basic concepts and diligently practicing on a variety of problems. An ability to skillfully combine like terms with exponents is, indeed, a powerful ally in your journey through mathematics.