AI Tutor
Imagine staring at 720\sqrt{720} during a timed math exam—do you build a 10-branch prime factor tree, or do you simplify it in five seconds flat?
Knowing the landmark squares up to 20220^2 turns tedious radical simplification into instant mental pattern recognition.

The 1 to 20 Benchmark Squares

A perfect square is an integer created by multiplying a whole number by itself, such as 12×12=14412 \times 12 = 144.
The number sitting directly inside a radical symbol is called the radicand.
Notice the digit reversal between 132=16913^2 = 169 and 142=19614^2 = 196, and remember that every square of a base ending in 55 always ends in 2525.
How do these twenty numbers unlock messy radical expressions without factoring all the way down to primes?

The Product Rule and Factor Hunting

The product property of radicals states that for non-negative numbers, a⋅b=a⋅b\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}.
This rule holds because squaring undoes a root: (ab)2=(a)2(b)2=ab(\sqrt{a}\sqrt{b})^2 = (\sqrt{a})^2(\sqrt{b})^2 = ab, which proves ab\sqrt{a}\sqrt{b} must be the square root of abab.
To simplify N\sqrt{N} in one move, you want to factor NN into its greatest perfect square factor and a leftover term.
Let's simplify 288\sqrt{288}.