91Ó°ÊÓ

The distributive property is a crucial algebraic rule that allows us to simplify expressions and perform multiplication in an expanded form. It states that for any three numbers a, b, and c, the equation \( a(b + c) = ab + ac \) holds true. This means that a single term outside the parentheses can be distributed or multiplied across each term inside the parentheses.

For instance, take the expression \(3h[2h + 5(h + 2)]\). Applying the distributive property, you first multiply the outer \(3h\) by \(2h\) and then by the entire expression \(5(h + 2)\). This results in \(3h \cdot 2h + 3h \cdot 5(h + 2)\), effectively breaking down the complex expression into simpler parts that can be further simplified.

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. When simplifying an expression, it's important to combine these like terms to reduce the expression to its simplest form.

For example, after distributing the \(3h\) across the expression in the problem above, we obtain terms \(6h^2\) and \(15h^2\), both of which are like terms since they have the variable \(h\) raised to the second power. By adding their coefficients, we get \(6 + 15\) for a combined term of \(21h^2\). It's essential to remember that only the coefficients are added or subtracted; the variable part remains unchanged. This step significantly clarifies the expression and brings us closer to the final simplified form.

Algebraic multiplication involves multiplying variables and their coefficients. It's a pivotal procedure when dealing with algebraic expressions. As per the standard rules of multiplication, you multiply the coefficients (numerical parts) and add the exponents of the variables if the variables are the same.

In our expression above, when we multiply \(3h\) by \(5h\), we multiply the coefficients (3 and 5) and add the exponents of \(h\), since they are similar, to get \(15h^2\). When multiplying a variable by a number, like \(3h \cdot 10\), we simply multiply the coefficient of the variable (3) by the number (10) to get \(30h\). Recognizing patterns in algebraic multiplication makes it easier to perform these operations quickly and accurately, leading to the complete simplification of the algebraic expression.