How to Multiply Expressions with Exponents: A Student's Guide

Multiplying expressions with exponents can seem challenging at first, but with the right approach and understanding of the rules, it becomes much simpler. Let's break it down step by step using the example of $(-5d^4)(5d^2)$.

### Step 1: Multiply the Coefficients
The first step in multiplying any expressions is to handle the coefficients, which are the numerical parts of the terms. In our example, we have -5 and 5. When we multiply these numbers:

-5 × 5 = -25

So, we replace the coefficients in our expression with -25.

### Step 2: Multiply the Variables with Exponents
Next, we will focus on the variable part of the expression, $d^4$ and $d^2$. When multiplying expressions that have the same base (in this case, the base is $d$), we use the rule of exponents that states:

\[ a^m \times a^n = a^{m+n} \]

Applying this rule to our variables:

- Start with $d^4 × d^2$.
- According to the exponent rule, we add the exponents: 4 + 2 = 6.
- Therefore, $d^4 × d^2 = d^6$.

### Final Answer
Combining both parts, we have:

\[ (-5d^4)(5d^2) = -25d^6 \]

### Real-World Application
Understanding how to multiply expressions with exponents is essential not just for math classes but also for various real-world applications. For example, in physics, the laws of motion often involve equations where variables are raised to powers. Similarly, in economics, formulas may include exponential terms to represent growth rates.

### Additional Tips
- Always check your work by reviewing each step to ensure accuracy.
- Practice with different examples to reinforce your understanding of multiplying expressions with exponents.

If you want to dive deeper into the rules for multiplying exponents or find more examples, check out other sections on Train Your Brain. Learning these concepts will not only help in your current studies but also in future mathematical challenges!