Powers, roots and indices
| English | Chinese | Pinyin |
|---|---|---|
| power | 幂 | mì |
| index | 指数 | zhǐ shù |
| base | 底数 | dǐ shù |
| square root | 平方根 | píng fāng gēn |
| cube root | 立方根 | lì fāng gēn |
| laws of indices | 指数定律 | zhǐ shù dìng lǜ |
| reciprocal | 倒数 | dào shǔ |
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 and roots lab
square = x^2
Change the base and see powers grow while roots undo powers.
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.
$5^2 \times \sqrt[3]{8} = 25 \times 2 = 50$. Evaluate the power and root first, then multiply.
Work out 5² × ∛8.
5² = 25 and ∛8 = 2, so 25 × 2 = 50.
The laws of indices 指数定律
- These three rules let you simplify any power expression:
Same base only. You can only add/subtract indices when the bases match. $2^3 \times 3^2$ does not simplify to $6^5$ — the bases are different.
Using the index laws, what is 2³ × 2⁴ as a single number?
2³ × 2⁴ = 2⁷ = 128.
Special powers (Extended)
- $a^0 = 1$ for any non-zero $a$ (even $7^0 = 1$).
- $a^{-n} = \dfrac{1}{a^n}$ — a negative index means a reciprocal 倒数 ($3^{-2} = \dfrac{1}{9}$).
- $a^{1/n} = \sqrt[n]{a}$ — fractional powers are roots ($8^{1/3} = \sqrt[3]{8} = 2$).
Evaluate 81^(1/2).
81^(1/2) = √81 = 9.
What does 3⁻² equal?
3⁻² = 1/3² = 1/9. A negative index means reciprocal, not negative.
For any non-zero number a, a⁰ = 1.
The zero-index rule gives a⁰ = 1 for all a ≠ 0.
Complete: 4⁻¹ = ______.
4⁻¹ = 1/4 = 0.25. A power of −1 gives the reciprocal.
You've got it
- multiply same-base powers → add indices; divide → subtract; power of a power → multiply
- $a^0 = 1$, $\;a^{-n} = \dfrac{1}{a^n}$, $\;a^{1/n} = \sqrt[n]{a}$
- $5^2 \times \sqrt[3]{8} = 25 \times 2 = 50$; $\;2^3 \times 2^4 = 2^7 = 128$