Question

The bold question I’m just confused

Answer 1

To show that $y = b^{x+h} = ab^x$, first use the property that a sum in an exponent can be written as product of bases with the separate parts as their exponents:

$b^{x+h} = b^x\cdot b^h$

from that, it is easy to see that, if x varies, but h does not, $b^h$ is constant and can be written as a number, say, "a", so

$y=b^{x+h} = b^x\cdot b^h=a\cdot b^x$

which completes the algebraic argument.