6 ms·
Raising a base to a power is a prescription for how many times to multiply by the base. If you first raise it to one exponent, m say, then to another exponent,
by wbhart 15y ago
Raising a base to a power is a prescription for how many times to multiply by the base. If you first raise it to one exponent, m say, then to another exponent, n say, and then multiply, you have first multiplied by the base m times, then multiplied by the same base a further n times. In total you have multiplied by the base m + n times.
Back in those days they would have used slide rules and understood logarithms very well (which they used for multiplication by adding logarithms, essentially). So they may have just answered that to multiply values is to add their logarithms and exponentiate. If the logarithm is taken to the common base, the logarithms are given by the respective exponents.
- Peaker 15y agoThat explanation is somewhat problematic with fractional exponents. The generalization of exponentiation makes the "multiply N times" explanation fail.
- nitrogen 15y agoNot really; a fractional exponent n yields the quantity one would have to multiply 1/n times to return the original value. Multiplication an integer number of times could be seen as a special case of a broader concept of "fractional" multiplication (much like the gamma function (Γ(n)) extends the discrete factorial to a continuous domain).
- Peaker 15y ago1/n times may not necessarily be whole either. How do you explain irrational exponents this way, for example? What about complex exponents? Indeed you can extend the special case to the continuous domain -- but then the definition is expanded as well. I still think "multiply N times" is just a special-case, and as such, not usable as a definition -- let alone an explanation of why we can add exponents in the general case.