Final answer:
A number and its opposite sum to 0, demonstrating they are additive inverses, exemplified by p + (-p) = 0. Real-world contexts include balancing a budget, where equal income and expense net to zero. Negative powers represent taking the reciprocal of the base to the positive power.
Step-by-step explanation:
To show that a number and its opposite have a sum of 0, illustrating that they are additive inverses, let's consider a number 'p' and its opposite '-p'. By definition, the sum of these two numbers is p + (-p). According to the rules of addition, when two numbers with opposite signs are added together, we subtract the smaller number from the larger one, and the answer will have the sign of the larger number. However, since both numbers are of the same magnitude but opposite signs, the result is zero. Mathematically, it is expressed as p + (-p) = 0.
For real-world contexts, adding rational numbers relates to situations like balancing a financial budget, where incomes (positive values) and expenses (negative values) affect one's net balance. When income (p) and expense (-p) are equal, the net balance is zero.
The concept of negative powers equates to taking the reciprocal of the base raised to the positive power. Therefore x-n = 1 / xn, as seen in the expression 3-4 equaling 1 / 34 or 1 / (3 × 3 × 3 × 3). This illustrates the 'add the exponents' rule where 34 × 3-4 = 30 = 1.