91Ó°ÊÓ

When simplifying algebraic expressions, the process of multiplying coefficients is foundational. Coefficients are the numerical parts of the terms in an expression. In our example, the coefficients are (-5) from (-5a^2b^2) and (6) from (6a^3b^6). To multiply these, simply treat them like ordinary numbers. So, when we multiply (-5) by (6), we get (-30).

This step is important because it determines the numerical part of our answer. Remember, the sign of the coefficient is also significant. Multiplying two numbers with different signs, as in our example, results in a negative number. Thus, understanding how to multiply coefficients is crucial to simplifying algebraic expressions correctly.

Another key element in simplifying algebraic expressions involves adding exponents of like terms. Like terms are terms that have the exact same variable raised to the exact same power. However, when we multiply like terms, we do not literally add the exponents, but apply the rule of exponent addition. This rule states that when you multiply two powers with the same base, you should add the exponents.

For instance, in the given expression (-5a^2b^2) * (6a^3b^6), both terms have the variable a with powers of 2 and 3 respectively, and variable b with powers of 2 and 6. According to the rule of exponent addition, we add the exponents for a resulting in a^{2+3} which simplifies to a^5, and for b we get b^{2+6} simplifying to b^8. Understanding how to correctly add exponents is essential to simplifying expressions without making common mistakes.

Algebraic expressions are a mix of numbers, variables, and arithmetic operations. Our example expression is (-5a^2b^2)(6a^3b^6), a product of two binomials. To simplify such expressions, you need to follow certain algebraic rules methodically. The first step is to multiply the coefficients, and the second step involves adding the exponents of like variables.

By performing these operations correctly, we arrive at the final simplified form of the algebraic expression. In our case, after multiplying coefficients and adding exponents, the complete simplified expression is -30a^5b^8. It's essential to be familiar with the basic rules governing algebraic expressions, which serve as the building blocks for more complex mathematics.