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Simplifying expressions is all about making algebraic expressions easier to work with. This often involves combining like terms, which means identifying terms that have the same variable and the same power. Once you find these terms, you can add or subtract them to make the expression simpler.
For instance, in our example expression \((x^{2}y^{5})(x^{2}z^{5})(-3y^{2}z^{3})\), we first need to bring all like terms together. This simplifies the repetition of multiplication by dealing with the coefficients and like variables collectively.
If you see identical variables with different exponents, you should consider the rules of exponents to combine them effectively. Simplifying expressions like these helps in reducing complex problems to simpler, more manageable forms.

Understanding and applying the rules of exponents is key when simplifying expressions that include powers. Let's break down some of these rules that you need to know:

• **Product of Powers**: When multiplying like bases, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• **Power of a Power**: When raising a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• **Power of a Product**: Distribute the exponent to each factor inside the parentheses: \((ab)^n = a^n \cdot b^n\).

In our example, using the product of powers rule, we combine like terms: \(-3x^{2+2}y^{5+2}z^{5+3}\). These rules help streamline calculations and simplify expressions efficiently.

Mathematical operations include the basic arithmetic functions like addition, subtraction, multiplication, and division. In algebra, these operations are extended to work with variables and expressions.
When simplifying an expression like \((-3x^{2}y^{5})(x^{2}z^{5})(y^{2}z^{3})\), focus on multiplication, which involves:

• Multiplying constants (coefficients such as -3)
• Applying multiplication to variables by using exponent rules

In this specific example, start by multiplying the constants and then deal with the multiplication of the variables. Ensure you apply the correct exponent rules to simplify each variable power. Finally, you'd calculate to get the expression \(-3x^{4}y^{7}z^{8}\), making it much simpler to handle. Breaking down such problems into smaller, manageable parts can significantly aid in understanding and efficiency.