Final answer:
The question is about algebraic techniques for evaluating linear expressions involving addition and subtraction of vectors and integer multiplication. The key principles involve considering vector direction and applying multiplication rules for integers and exponentials.
Step-by-step explanation:
The student is asking about the addition and subtraction of vectors, as well as the rules for multiplying integers, both of which are important concepts in mathematics. When adding or subtracting vectors that act in a straight line, we can apply algebraic techniques similar to those used when adding or subtracting scalar quantities (ordinary numbers). A key principle is that the direction of the vector must be considered; for instance, a vector acting to the left is considered negative, while one acting to the right is positive.
For multiplying integers:
-
- Two positive numbers multiplied result in a positive product (e.g., 2x3 = 6).
-
- Two negative numbers result in a positive product (e.g., (-4) x (-3) = 12).
-
- A positive number and a negative number multiplied result in a negative product (e.g., (-3) x 2 = -6).
These rules also apply to division because division can be considered the inverse operation of multiplication. The commutative property of addition, which states that numbers can be added in any order without changing the result, also applies to multiplying or dividing numbers in a sequence.
When dealing with exponential terms:
-
- Multiply the digit terms normally.
-
- Add the exponents when multiplying exponential terms to combine them into a single term.
-
- The exponent zero applied to any base (except zero) results in the value of one.
Finally, to add or subtract fractions, use your intuition by finding a common denominator and applying the principles of addition and subtraction to the numerators while keeping the common denominator the same.