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In algebra, exponents represent how many times a number, the base, is multiplied by itself. An exponent is usually written as a superscript to the right of the base. For instance, in the expression \(7x^{4}y^{5}\), 7 is the coefficient, \(x\) and \(y\) are the bases, and 4 and 5 are their respective exponents. Understanding the relationship between bases and exponents is crucial when performing operations like multiplication of polynomials.

In the problem given, we have two polynomial terms, or monomials, that we need to multiply. Each term consists of a coefficient followed by variables as bases raised to certain powers as exponents. The rule for multiplying these terms together involves handling the coefficients separately from the bases with their exponents, which we will address in the steps below.

A monomial is a type of polynomial with just one term consisting of a coefficient and one or more variables raised to exponents. The monomials in our example are \(7x^{4}y^{5}\) and \( -10x^{7}y^{11}\). When we multiply monomials, we multiply the coefficients (the numbers in front of the variables) together and apply the rules of exponents for the variables with the same base.

Multiplying Monomials

To illustrate, if you multiply these two monomials, you begin by multiplying the coefficients 7 and -10 to get -70. Then, for each variable that appears in both monomials, you add their exponents to find the new exponent for that variable in the product. This process simplifies the multiplication of variables raised to powers and is an efficient and systematic approach for multiplying any number of monomials together.

The rule of exponents that applies here is the product of powers rule, which states that when you multiply two powers with the same base, you add their exponents. For example, \(x^a \cdot x^b = x^{a+b}\).

Applying this rule to our problem, after multiplying the coefficients, we get -70. Then we add the exponents of matching variables: for \(x\), we add 4 (from \(x^{4}\)) and 7 (from \(x^{7}\)) to get 11, and for \(y\), we add 5 (from \(y^{5}\)) and 11 (from \(y^{11}\)) to get 16. Therefore, our final answer is \( -70x^{11}y^{16}\), a single monomial. Understanding this rule is the key to successfully multiplying polynomials, and it's a foundational concept that will recur in more complex algebraic operations.