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When multiplying monomials, one of the first steps is multiplying the coefficients. Coefficients are the numerical part of terms that contain variables. In the provided example, we have two coefficients, 2 and 5, associated with variable 'n'.

To multiply the coefficients, simply multiply the numbers as you would in a regular arithmetic operation. It’s just like taking 2 apples and multiplying them by 5, giving you a total of 10 apples. It’s direct and straightforward. Here’s how you do it mathematically:

• Identify the coefficients: 2 and 5.
• Multiply them together: 2 * 5.
• The result is 10.

Remember, this is an essential step in simplifying algebraic expressions because it sets the stage for combining like terms and simplifying the expression as a whole.

Once the coefficients are multiplied, the next step in multiplying monomials is to handle the variables with exponents. If the same variable is raised to different powers, we add the exponents together. This is based on the laws of exponents, which tell us that when you multiply like bases you add the exponents.

For example, considering the variable term 'n' raised to the 3rd and 4th power, as in our exercise \( n^3 \cdot n^4 \) we apply the rule:

• Identify the exponents: 3 and 4 for the variable 'n'.
• Add the exponents together: 3 + 4.
• The new exponent of 'n' is 7.

The variable 'n' raised to the 7th power (\( n^7 \) ) represents the combined growth rate or repeated multiplication of 'n' due to its original individual powers. This step retains the integrity of exponential growth or decay that the variable is subject to in different monomials.

After we've multiplied the coefficients and added the exponents, we can compile our findings into a single algebraic expression. In the context of our example, combining the product of the coefficients (10) with the new exponent of the variable 'n' (7) results in the final algebraic expression \( 10n^7 \).

Essence of Simplifying Algebraic Expressions

Algebraic expressions are meant to be as simple as possible to make calculations easy to perform and understand. This simplicity is achieved by combining like terms and using the laws of arithmetic and exponents.

Here’s a quick summary of the process to create a simplified algebraic expression:

• Multiply the coefficients.
• Add the exponents of the same variables.
• Combine the multiplied coefficient with the variable raised to the new exponent.

Remember, the goal is to obtain a clear, concise, and accurate representation of quantities that shows the relationship between variables and constants in algebraic form.