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Exponent rules help us make sense of expressions with powers. They tell us how to handle the exponents when we multiply, divide, or raise one power to another. Some key exponent rules are:

• Product of Powers Rule: When multiplying two expressions with the same base, add their exponents: \ a^m × a^n = a^{m+n}.
• Power of a Power Rule: When raising an exponent to another exponent, multiply the exponents: \ (a^m)^n = a^{m \times n}.
• Power of a Product Rule: To raise a product to a power, apply the exponent to each factor in the product: \ (ab)^m = a^m \times b^m.

In our exercise, we used the Power of a Product and Power of a Power rules. We started with \( f(x) = (4e^x)^{3x} \) and applied the rules to break it down into simpler terms.

Simplifying expressions means making them easier to work with. We used exponent rules to break down the complex expression \( (4e^x)^{3x} \). This involved applying the Power of a Product Rule: \( (ab)^m = a^m \times b^m \). We treated \( 4 \ \text{and} \ e^x \) as factors and the exponent \( 3x \) applied to both. So, \( (4e^x)^{3x} \) became \( 4^{3x} \times (e^x)^{3x} \).
Next, we simplified further using the Power of a Power Rule: \( (a^m)^n = a^{m \times n} \). Applying it to \( (e^x)^{3x} \), we got \( e^{3x^2} \).
Combining everything, we reached our final expression: \( f(x) = 4^{3x} \times e^{3x^2} \).

Exponential functions, such as \( e^x \), are functions where the variable is in the exponent. They show growth or decay at an increasingly rapid rate. The base of natural exponential functions is the constant \( e \), approximately 2.718.
Exponential functions are powerful tools in calculus. They're used in various applications, like compound interest, population growth and decay, and many forms of natural phenomena.
In our exercise, we dealt with \( e^x \), highlighting how exponent rules simplify exponential functions. Understanding these rules allows us to manipulate and simplify these functions effectively.
When working with exponential functions:

• Look for opportunities to apply exponent rules to simplify the terms.
• Combine like terms whenever possible.
• Always pay attention to the base of the exponential expressions.

This helps in maintaining clarity and accuracy through more complex calculus problems.