Simplify √a to b√c: pull out the largest perfect square

The magic rule

$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$

If $a$ is a perfect square, pull it out: $\sqrt{a^2 \times b} = a \sqrt{b}$.

Method

1. Decompose $n$ under the root.
2. Spot the largest perfect square among the divisors.
3. Pull out its root.

Perfect squares to know

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

Typical examples

$\sqrt{72}$: $72 = 36 \times 2$. $\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$.

$\sqrt{50}$: $50 = 25 \times 2$. $\sqrt{50} = 5\sqrt{2}$.

$\sqrt{288}$: $288 = 144 \times 2$. $\sqrt{288} = 12\sqrt{2}$.

$\sqrt{300}$: $300 = 100 \times 3$. $\sqrt{300} = 10\sqrt{3}$.

Trick: prime factorization

For $\sqrt{n}$, factor $n$ into primes. Each pair of a same factor comes out of the root (as a square).

Ex: $\sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5}$.

Operations with simplified roots

Add/subtract only possible if same part under the root: $3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2}$.

But $3\sqrt{2} + 5\sqrt{3}$ can’t be simplified further.

Common mistakes

1. Pulling out a non-square factor: only perfect squares come out.
2. $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ (WRONG).
3. Not pulling out the largest square: $\sqrt{20} = 2\sqrt{5}$ (since $20 = 4 \times 5$), fully simplified.

?À toi de jouer

Simplify √98?

A.7√2B.√98C.2√49D.7√14

Valider