91Ó°ÊÓ

To start with, let's talk about the Distributive Property. This is a fundamental concept in algebra that helps in simplifying expressions and solving equations. The Distributive Property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. So, if you have an expression like \(a(b + c)\), you can distribute \(a\) to both \(b\) and \(c\) to get \(ab + ac\).

In our exercise, we are given 7\(m^2(2m^4 - 3m + 4)\). Here, 7\(m^2\) is outside the parentheses and will be distributed to each term inside the parentheses: \(2m^4\), \(-3m\), and \(4\). This means:

• \(7\(m^2 × 2m^4\)\)
• \(7\(m^2 × (-3m)\)\)
• \(7\(m^2 × 4\)\)

Distributing each multiplication helps simplify the expression step by step.

Exponents can sometimes be a bit tricky, but once you get the hang of the rules, they become much easier to manage. When you're multiplying terms with exponents, you simply add the exponents if the base is the same.

For example, in the expression \(7m^2\) and \(2m^4\), both terms have the same base \(m\). So, you add the exponents: \(2 + 4 = 6\). Therefore, \(7m^2 × 2m^4\) becomes \(14m^6\).

The same rule applies when you multiply \(7m^2\) with \(-3m\): the exponents of the \(m\) terms are \(2\) and \(1\), respectively. Adding these exponents gives us \(2 + 1 = 3\), making the term \(-21m^3\).

• When multiplying terms, add the exponents: \(a^m × a^n = a^{m+n}\)

This rule makes it easy to simplify terms that have exponents.

The final step in our process is to combine all the terms into one simplified expression. After distributing and applying the exponent rules, we have:

• \(14m^6\)
• \(-21m^3\)
• \(28m^2\)

Combining these, we get the simplified expression: \(14m^6 - 21m^3 + 28m^2\). Always write your final expression in descending order of exponents, starting with the highest one, which is exactly what we have done here.

Simplifying expressions helps in making the terms easier to manage and understand, which is particularly helpful when moving on to more complex algebraic operations.