Radicals December 6,2024

A radical is a symbol that represents the root of a number, most commonly the square root. For example, $$\text{√}x$$ represents the square root of x.

You can only combine $$\text{√}a + \text{√}b$$ if a and b are perfect squares or if they have a common factor.

To simplify $$\text{√}50$$ , factor it into $$\text{√}(25 \times 2) = \text{√}25 \times \text{√}2 = 5\text{√}2$$ .

$$\text{√}300 + \text{√}75 = \text{√}(100 \times 3) + \text{√}(25 \times 3) = 10\text{√}3 + 5\text{√}3 = 15\text{√}3$$ .

Like radicals can be combined by adding or subtracting their coefficients. For example, $$3\text{√}2 + 5\text{√}2 = (3 + 5)\text{√}2 = 8\text{√}2$$ .

$$2\text{√}27 + 3\text{√}3 = 2\text{√}(9 \times 3) + 3\text{√}3 = 6\text{√}3 + 3\text{√}3 = 9\text{√}3$$ .

$$5\text{√}75 + 8\text{√}27 = 5\text{√}(25 \times 3) + 8\text{√}(9 \times 3) = 25\text{√}3 + 24\text{√}3 = 49\text{√}3$$ .

Rationalizing the denominator means to eliminate any radicals in the denominator of a fraction by multiplying the numerator and denominator by a suitable radical.

$$\text{√}32 = \text{√}(16 \times 2) = 4\text{√}2$$ .

$$3\text{√}50 - 5\text{√}32 = 3(5\text{√}2) - 5(4\text{√}2) = 15\text{√}2 - 20\text{√}2 = -5\text{√}2$$ .