Final answer:
In multiplication, the sign of the product depends on the signs of the numbers involved: two positives or two negatives result in a positive product, while a positive and a negative give a negative product. Raising a number to a power represents repeated multiplication, and multiplication by a scalar affects a vector's magnitude but not its direction.
Step-by-step explanation:
A number that is multiplied by another number to get a product is fundamental to the concept of multiplication in mathematics. This operation is used across a wide variety of math topics from basic arithmetic to more advanced topics like algebra and vectors.
When two positive numbers are multiplied, for instance 2x3, the result is a positive number, which in this case is 6. Just like when two negative numbers, such as (-4) x (-3), are multiplied, the outcome is again a positive number, 12. This demonstrates that the sign of the product is dependent on the signs of the factors being multiplied. The multiplication of numbers with opposite signs will always result in a negative product, for example, (-3) x 2 equals -6, and 4 x (-4) equals -16.
Division follows similar rules to multiplication with regard to the sign of the result. A positive number divided by a positive number remains positive, a negative divided by a negative is positive, and a positive divided by a negative or a negative divided by a positive yields a negative result.
The concept of raising a number to a power, such as 4³, which means 4 x 4 x 4, is just a shorthand for repeated multiplication. The opposite operation, or the 'inversion' of multiplication, involves the use of negative exponents. For instance, 3⁻⁴ is equivalent to 1/3⁴, or 1/(3 x 3 x 3 x 3).
In more advanced mathematics, multiplication applies to vectors as well. When a vector is multiplied by a positive scalar, similar to a number being multiplied by another number, the product is a new vector, scaled in size but parallel in direction to the original. This is distinct from the scalar and vector products within vector multiplication, which yield different types of products based on their respective definitions and properties.