🟩 平方根：铺满正方形的那条边🟩 Square Roots: the side that fills the square
上一片用 10 的幂把大数写短。这一片反过来问：哪个数自己乘自己，刚好得到它？小算盘机器人在铺地砖时，就撞上了这道反推题。Last leaf used powers of ten to shrink big numbers. This leaf flips the question: which number times itself gives exactly this? The little abacus robot runs right into it while tiling a floor.
第 1 步：把正方形铺满Step 1: Fill the square
先看看机器人手上在忙什么。First, a look at what the robot is busy with.
方砖被摆成 6 行 6 列，正好填满一块正方形地面。数一数：每行 6 块，一共 6 行，6 × 6 = 36 块。现在把问题反过来：如果面积是 36，边长是多少？答案是 6——因为 6 自乘正好得 36。The tiles sit in 6 rows of 6, filling one square floor exactly. Count them: 6 per row, 6 rows, so 6 × 6 = 36 tiles. Now flip the question: if the area is 36, what is the side? It is 6 — because 6 times itself is exactly 36.
那么，这道“反过来的题”有没有名字？So does this “backwards question” have a name?
有，它叫平方根。6 自乘得 36，就说 6 是 36 的平方根，写成 √36 = 6。规则只有一条：面积 = 边长 × 边长；反过来读，边长 = 面积开平方。49 的平方根是 7，81 的平方根是 9；当地砖边长用时，只取正数。Yes — the square root. Since 6 times itself gives 36, 6 is the square root of 36, written √36 = 6. One rule does it all: area = side × side; read backwards, side = the square root of the area. The square root of 49 is 7, of 81 is 9 — and as a floor side, we take the positive one.
如果面积不是平方数，问号还能解出来吗？What if the area is not a perfect square — can the question mark still be solved?
第 3 步：把问号解出来Step 3: solve for the question mark
把题目写成 ? × ? = 36，问号就是答案（6）。可要是面积是 40 呢？6 × 6 = 36、7 × 7 = 49，都不等于 40，所以 √40 夹在 6 和 7 之间，怎么乘都乘不出来，约等于 6.32。平方数开出整数，其余开出无限小数——它们都是平方根。Write it as ? × ? = 36 — the question mark is the answer (6). But what about area 40? 6 × 6 = 36 and 7 × 7 = 49, neither is 40, so √40 sits between 6 and 7 and never comes out clean: about 6.32. Perfect squares give whole numbers, the rest give endless decimals — all of them are square roots.
🎮 你来铺地砖（30 秒）🎮 Your turn: tile the floor (30 seconds)
拖滑块改变边长，让正方形面积恰好等于目标值。铺对 4 块就通关。Drag the slider to change the side until the square's area hits the target. Get all 4 right to win.
一句话记住它：平方根就是“哪个数自乘得它”——已知面积，反推出边长。Remember it in one line: a square root answers “which number times itself gives this?” — from area back to side.
√36 = 6，因为 6 × 6 = 36；反推题写成 ? × ? = 面积√36 = 6 because 6 × 6 = 36 — the backwards question is ? × ? = area面积 = 边长 × 边长；平方根把面积变回边长area = side × side; the square root turns area back into side边长只取正数；√40 ≈ 6.32 这种开不尽的数也是平方根sides are positive; endless decimals like √40 ≈ 6.32 are square roots too
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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