91Ó°ÊÓ

Algebraic expressions are combinations of variables, numbers, and mathematical operations such as addition, subtraction, multiplication, and division. These expressions have no equal sign, distinguishing them from equations.
For instance, in the expression \((-12x)(3y)\left(-\frac{3}{4}xy^6\right)\), we have three distinct terms.
- The first term, \(-12x\), includes a coefficient of -12 and the variable \(x\).- The second term, \(3y\), has a coefficient of 3 with the variable \(y\).- The third term, \(-\frac{3}{4}xy^6\), consists of the coefficient \(-\frac{3}{4}\) alongside the variables \(x\) and \(y^6\).
These components can be rearranged or combined according to specific algebraic rules, helping us to manipulate and solve them for different variables or simplify them into a more understandable form.

Polynomial operations involve performing mathematical processes, such as addition, subtraction, and multiplication, on polynomials. A polynomial is an algebraic expression made up of terms where each term includes a variable raised to a whole number exponent.
When multiplying polynomials, like in our exercise, we follow these simple rules:

• Multiply the coefficients, which are the numbers in front of the variables.
• Multiply the variables, ensuring to apply the laws of exponents correctly.

In our case, we first multiply the coefficients: \((-12)\), \(3\), and \(-\frac{3}{4}\), resulting in \(-27\).
Next, we multiply the same base variables:
- \(x\) terms: \(x \times x = x^2\)
- \(y\) terms: \(y \times y^6 = y^{1+6} = y^7\)
The final product is then written as \(-27x^2y^7\). This approach helps in simplifying and combining polynomial expressions efficiently.

Exponents are a mathematical way to represent repeated multiplication of a number by itself. They appear in algebraic expressions, indicating how many times the base (number or variable) is multiplied by itself.
Consider the variable products in the example \(x\) and \(y\).
When multiplying variables with exponents, the key rule is to add the exponents if the bases are the same. Here’s how it works:

• Multiplying \(x\) terms: \(x\times x = x^{1+1} = x^2\)
• Multiplying \(y\) terms: \(y \times y^6 = y^{1+6} = y^7\)

Understanding how exponents work is crucial, especially in polynomial operations, as it dictates how terms are combined and simplified.
This helps significantly in cases where algebraic expressions involve multiple terms with exponents, allowing us to efficiently arrive at simplified results, such as the final answer: \(-27x^2y^7\).