指数と標準形

For powers with the same base, multiply by adding the indices and divide by subtracting them: $a^m\times a^n=a^{m+n}$ and $a^m÷a^n=a^{m-n}$. A power raised to a power multiplies the indices: $(a^m{)}^n=a^{mn}$. For example $x^3\times x^4=x^7$, $x^9÷x^2=x^7$, and $(x^2{)}^5=x^{10}$.

Any nonzero number to the power $0$ equals $1$: $a^0=1$. A negative index means a reciprocal: $a^{-n}=\frac{1}{a^n}$. A fractional index means a root, so $a^{1/2}=\sqrt{a}$ and more generally $a^{m/n}=\sqrt[n]{a^m}$. For example $5^0=1$, $2^{-3}=\frac{1}{8}$, and $9^{1/2}=3$.

Standard form (scientific notation) writes a number as $a\times 10^n$, where $1\le a<10$ and $n$ is an integer. Large numbers give a positive $n$; small numbers give a negative $n$. For example $52000=5.2\times 10^4$ and $0.0007=7\times 10^{-4}$. The key is to place the decimal point so that $a$ is at least $1$ but less than $10$.

To compute with numbers in standard form, handle the $a$ parts and the $10^n$ parts separately: $(a\times 10^m)\times (b\times 10^n)=(a\times b)\times 10^{m+n}$, and subtract the indices when dividing. Finally adjust so the answer has $1\le a<10$. For example $(3\times 10^4)\times (2\times 10^5)=6\times 10^9$ and $(8\times 10^6)÷(2\times 10^2)=4\times 10^4$.