🌱Square Roots

You have long known that $5^2 = 25$. Now flip the question around: which number times itself gives 25? The answer is still 5 — and asking it backwards is exactly what a square root does. Squares and cubes taught you to square a number; today we learn to undo the squaring.

The undo button for squaring

The symbol $\sqrt{25}$ reads "the square root of 25" and asks: who, squared, equals 25?

$\sqrt{25} = 5$

The reason is short: $5^2 = 25$. Squaring presses a number flat; taking the square root restores the photo. The two are inverse operations, just like addition and subtraction, or multiplication and division.

The perfect-square roster

Numbers that come out perfectly clean are called perfect squares. Memorize the squares of 1 through 15, and taking a root becomes "spotting a friend on the roster":

$1,\ 4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100,\ 121,\ 144,\ 169,\ 196,\ 225$

For instance $\sqrt{144} = 12$, because 12 is right there on the list.

Squeezing the in-between ones

What if the number is not on the roster? Take $\sqrt{50}$ and squeeze it with multiplication:

$7^2 = 49 < 50 < 64 = 8^2$

So $\sqrt{50}$ is stuck between 7 and 8, and very close to 7 (50 is only a hair above 49) — estimate it as about $7.07$. No calculator needed: the roster plus a squeeze pins it down to two decimal places.

4 × 4 = ?

✓

🔥 0 · ★ 0

Multiplication is the hidden engine of roots. Drill the products up to 15 until they are automatic, and 50 will instantly remind you that $7 \times 7 = 49$ — your estimates get much faster.

Irrational numbers step on stage

The squeeze cannot always finish the job: $\sqrt{2} \approx 1.414\ldots$ goes on forever, and it is not any fraction. Legend says the Pythagoreans in ancient Greece were shocked to discover this — such numbers are called irrational: perfectly real, but refusing to be written as a fraction.

Cube roots and calculators

The same story has a cube-flavored version: $\sqrt[3]{27} = 3$ because $3^3 = 27$, and $\sqrt[3]{64} = 4$ because $4^3 = 64$. For anything fancier, hand the work to a calculator — but estimate a rough range first and you will instantly see whether the machine made a mistake. The machine supplies digits; you supply the sanity check.

Check yourself

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Quick quiz

↺ Try again

1. 1. What is √81?

   940.58

2. 2. √50 sits between which two whole numbers?

   6 and 77 and 88 and 9

3. 3. What is ∛64?

   48about 21.3