Indices in algebra

The tower of powers

• How many times can you double a grain of rice before it fills a room? About 40 doublings — that's $2^{40} \approx 10^{12}$ grains.
• Index laws let you tame these enormous numbers, working with the powers instead of computing the values.

The three core rules

• Multiply same-base powers → add indices 指数: $\;a^m \times a^n = a^{m+n}$.
• Divide same-base powers → subtract indices: $\;a^m \div a^n = a^{m-n}$.
• Power of a power → multiply indices: $\;(a^m)^n = a^{mn}$.

Working with coefficients

• Separate the number part from the letter part, simplify each, then recombine.
• $12a^5 \div 3a^{-2}$: numbers $12 \div 3 = 4$; letters $a^5 \div a^{-2} = a^{5-(-2)} = a^7$. Answer: $4a^7$.
• $6x^7y^4 \times 5x^{-5}y = 30x^{7+(-5)}y^{4+1} = 30x^2y^5$.

Index equations 指数方程

• Write both sides with the same base, then equate the powers.
• Solve $2^x = 32$: since $32 = 2^5$, we get $x = 5$.
• Solve $3^{2x} = 81$: since $81 = 3^4$, $2x = 4$, so $x = 2$.

Where index laws appear

• Scientific notation relies on $10^a \times 10^b = 10^{a+b}$.
• Compound interest uses $(1+r)^n$ — a power of a real-world multiplier.
• Computer science: memory sizes are powers of 2 (KB $= 2^{10}$, MB $= 2^{20}$, GB $= 2^{30}$).

Powers and roots are inverses: the area of a square is $s^2$, and its side is $\sqrt{\text{area}}$.

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