91Ó°ÊÓ

Understanding exponent rules is crucial when simplifying algebraic expressions involving powers. When you have an expression like \( x^m \cdot x^n \) where \( x \) represents the base and \( m \) and \( n \) are the exponents, you simply add the exponents to multiply the bases. This is because you're multiplying \( x \) by itself \( m \) times, and then \( n \) times more.

For example, the expression \( x^3 \cdot x^2 \) simplifies to \( x^{3+2} = x^5 \). Similarly, when you have different bases with the same exponent, such as \( x^n \cdot y^n \) , the exponent rule doesn't apply directly because the bases are different. However, if the expression involves the same base but with different exponents as in the given problem with \( y \) (\( y^4 \cdot y^3 = y^{4+3} = y^7 \) ), the bases are multiplied and the exponents added.

Multiplying polynomials involves distributing each term in the first polynomial by each term in the second polynomial. When simplifying the expression \( (7xy^4)(-2xy^3) \), you deal with a simpler case of polynomial multiplication since each polynomial consists of a single term - these are known as monomials.

The process is straightforward: multiply the coefficients (numbers in front of the variables), and use the exponent rules for the variables. If the polynomials had more terms, you'd use the FOIL method (First, Outside, Inside, Last) to ensure each term in the first polynomial is multiplied by each term in the second polynomial, resulting in a polynomial of higher degree.

To simplify an algebraic expression, you often need to combine like terms. Like terms are terms that have exactly the same variables raised to the same powers, even though the coefficients might be different. For instance, \( 2x \) and \( 3x \) are like terms, and they combine to \( 5x \). But \( 2x \) and \( 2x^2 \) are not like terms because the exponents on \( x \) are different.

In the context of our example, \( -14x \) is combined with \( x^2y^7 \) by applying the exponent rules to \( x \) and \( y \) separately, resulting in the simplified expression \( -14x^2y^7 \) which doesn't have any like terms to combine further. Thus, combining like terms is essential to simplify and reduce an expression to its most concise form.