Powers, roots and indices

The chessboard grain problem

• A king offered a reward for the invention of chess. The inventor asked for 1 grain of rice on the first square, 2 on the second, 4 on the third, doubling each time.
• By the 64th square: $2^{63}$ grains — more rice than has ever existed on Earth.
• That's the power 幂 of indices — repeated multiplication grows astonishingly fast.

Powers

• A power (or index 指数) shows repeated multiplication: $2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$.
• The base 底数 is the number being multiplied ($2$); the index (or exponent) tells you how many times ($5$).
• Know your squares up to $15^2 = 225$ and cubes up to $5^3 = 125$.

Roots

• A square root 平方根 reverses squaring: $\sqrt{169} = 13$ (because $13^2 = 169$).
• A cube root 立方根 reverses cubing: $\sqrt[3]{8} = 2$ (because $2^3 = 8$).
• Roots and powers are inverse operations.

The diagonal of a square is the square root of twice the side squared — geometry gives us roots.

The laws of indices 指数定律

• These three rules let you simplify any power expression:

Zero, negative and fractional powers

• $a^0 = 1$ for any non-zero $a$ (even $7^0 = 1$).
• $a^{-n} = \dfrac{1}{a^n}$ — a negative index means a reciprocal 倒数 ($3^{-2} = \dfrac{1}{9}$).
• Extended: $a^{1/n} = \sqrt[n]{a}$ — fractional powers are roots ($8^{1/3} = \sqrt[3]{8} = 2$).

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