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In algebra, "like terms" refer to terms that have the same variables raised to the same power, even if their coefficients are different. Recognizing like terms is essential when simplifying expressions and equations.

When multiplying polynomials, it's crucial to group and multiply like terms together. This helps to simplify the expression and achieve the correct result. For example, if you're multiplying \((-x^3y^2)(xy^3)\), you want to identify terms that share the same variable base:

• Terms with the base \(x\): \(-x^3\) and \(x\).
• Terms with the base \(y\): \(y^2\) and \(y^3\).

Understanding and correctly identifying like terms is the first step in the process of polynomial multiplication. This recognition allows you to effectively apply the rules of exponents to simplify the expression.

Exponents are a fundamental part of algebra that denote repeated multiplication of a base. When multiplying terms with the same base, the exponents are added. This is known as the **product of powers property**.

Consider the terms \(-x^3\) and \(x\). Applying the product of powers property: \[-x^3 \times x = -x^{3+1} = -x^4.\]Similarly, for terms with the base \(y\), \[y^2 \times y^3 = y^{2+3} = y^5.\]This property streamlines the process of polynomial multiplication by reducing multiple steps into one concise operation. Keep in mind that this rule applies only to like terms with the same base. Understanding exponents and their rules is key to mastering algebraic expressions.

Algebraic expressions consist of variables, coefficients, and arithmetic operations. They form the basis of algebra and can be manipulated using various rules and properties.

In our example, the expression is \((-x^3y^2)(xy^3)\). To solve it, you need to:

• Identify and group like terms, as discussed previously.
• Utilize the rules of exponents to multiply the terms.
• Combine the results into a single expression: \(-x^4y^5\).

Algebraic expressions allow for a wide range of mathematical operations and problem-solving techniques. By understanding how to manipulate these expressions, you gain valuable skills for tackling more complex algebraic problems in the future. Mastery of terms, exponents, and algebraic rules opens the door to more advanced mathematical concepts and applications.