"Why" questions

why is a^-1 = 1/a

using laws of exponents: a^-1=a^(3-4) = a^3/a^4= a×a×a / a×a×a×a= 1/a

Why is a^0 = 1?

using laws of exponents: example 1= 125/125 =5^3/ 5^3 = 5^(3-3)= 5^0

Why can't we divide by zero

13÷0=c↔ 13= c×0 c would be undefined because no number multiplied by zero will equal anything but zero 0÷0= indeterminate because its not unique

why do two negatives make a positive?

In terms of money: you owe negative 5 people negative 10 dollars. five people need to give you ten dollars, you made 50 dollars! Definition of the multiplication of integers, can use number line to show as well

why do we line up the decimal places when adding/subtracting decimal numbers?

adding up like parts of a whole. maintaining place value

why does invert and multiply work when we divide fractions?

divide two fractions then use the fundamental law of arithmetic and multiply by a convenient form of zero. the denominator should then be itself multiplied by its reciprocal, which by the definition of multiplicative inverse of rational numbers, would be one. so what is left is the original numerator multiplied by the reciprocal of the original denominator. :)

why do we move the decimal point when multiplying decimals?

make the numbers into mixed numbers, then make the two sets of mixed numbers into improper fractions (ex 1.5→1 5/10→ 10/10 +5/10) there should be a power of ten on the bottom. say it is over 10,000 (dividing the numerator by 10,000), that would be the equivalent of moving the decimal place over 4 places.

Why do we have common denominators when adding/subtracting fractions?

we rewrite to have like parts of the same whole (denominator), then we add/subtract how many parts (numerator)