Identity

$ab = a b$ and $a / b = a / b$

Algebra • Surds • Simplification

Formula

ab = a b, \frac{a}{b} = \frac{a}{b}

For

a \geq 0

and

b \geq 0

ab = a b

; and for

a \geq 0

b > 0

\frac{a}{b} = \frac{a}{b}

. Both require the radicands to be non-negative.

How to use it

1Split a radicand into a perfect square times a remainder, then take the square out: $72 = 362 = 62$ .
2Take the root of a fraction by rooting the numerator and the denominator separately, once both are perfect squares.
3Run it in reverse to combine two roots before simplifying: $82 = 16 = 4$ .

Simplify

\frac{81}{49} \times \frac{196}{9}

Step-by-step

1
$\frac{81}{49} = \frac{81}{49} = \frac{9}{7}$
Why: Root the numerator and the denominator separately.
2
$\frac{196}{9} = \frac{196}{9} = \frac{14}{3}$
3
$\frac{9}{7} \times \frac{14}{3} = 6$

Answer

6

Watch out

There is no such law for sums:

a + b \neq = a + b

in general. The product law also fails for two negative radicands, where

- 1 - 1 \neq = 1

Builds on

Also known as

surd product law, surd quotient law, root of a fraction, square root of a quotient, split a square root