Multiplying and Dividing Radicals

A radical is an expression that includes a root, such as a square root (√) or cube root (∛).

To multiply two radicals, multiply the numbers outside the radicals and the numbers inside the radicals separately. For example, $$\text{If } a = \text{√}x \text{ and } b = \text{√}y, \text{ then } a \times b = \text{√}(xy)$$ .

$$6\text{√}2 \times 5\text{√}14 = 30\text{√}(2 \times 14) = 30\text{√}28 = 60\text{√}7$$ .

Multiply the numerator and denominator by $$\text{√}5$$ to rationalize: $$\frac{10\text{√}2 \times \text{√}5}{3\text{√}5 \times \text{√}5} = \frac{10\text{√}10}{15} = \frac{2\text{√}10}{3}$$ .

$$-2\text{√}7 \times 3\text{√}28 = -6\text{√}(7 \times 28) = -6\text{√}196 = -84$$ .

$$\frac{3\text{√}12}{\text{√}25} = \frac{3\text{√}(4 \times 3)}{5} = \frac{6\text{√}3}{5} = \frac{6\text{√}3}{5}$$ .

$$-2\text{√}10 \times 3\text{√}20 = -6\text{√}(10 \times 20) = -6\text{√}200 = -60\text{√}2$$ .

The square root of 25 is 5, since $$5^2 = 25$$ .

Rationalizing the denominator means to eliminate any radicals in the denominator of a fraction.

$$\text{√}12 = \text{√}(4 \times 3) = 2\text{√}3$$ .