This is a really basic question, but I have just got interested in math and learning rules about powers/indices and this confused me a little.
$\dfrac{a^3}{a^7}$ after they cancel out we get $\dfrac{1}{a^4}$
My question is why do we have 1 remain at the top? Why not 0, any simple explanation for this?
On
The reason is because when you have an expression like that, what you're effectively doing is $$\frac{a^3}{a^7} \div \frac{a^3}{a^3} = \frac{a^3\div a^3}{a^7\div a^3}$$ Where $\dfrac{a}{a}=1$, and $\dfrac{a^3}{a^3} = \dfrac{a}{a}*\dfrac{a}{a}*\dfrac{a}{a}$
On
$$\frac{a^3}{a^7}=\frac{a\cdot a\cdot a}{a \cdot a \cdot a \cdot a \cdot a\cdot a\cdot a}=\frac{1\cdot a\cdot a\cdot a}{a \cdot a \cdot a \cdot a \cdot a\cdot a\cdot a}=\frac{1\cdot a^3}{a^3\cdot a^4}=\frac{1}{a^4}$$
On
Because $1$ is the multiplicative identity not $0$
You don't 'cancel' numerator and denominator, you multiply numerator and denominator by the inverse of the common element in numerator and denominator and in $\frac{a^3}{a^7}$, the common part is $a^3$ itself , so when you multiply numerator and denominator by inverse of $a^3$, you get the multiplicative identity($1$ in this case) on top and $a^4$ in denominator.
"Cancelling" is a misleading expression. You're not cancelling anything, that is, just making something disappear. You're dividing the numerator and denominator by the same thing, in this case, $a^3$. In the numerator, you're dividing $a^3$ by $a^3$, giving 1. In the denominator, you're again dividing by $a^3$. Here you wind up with 3 sets of (a/a), which each turn into 1. and four a's that give the $a^4$