4E Adding Subtracting Multiplying Radicals

A radical is an expression that includes a root, such as a square root (√) or cube root (∛). For example, √4 is a radical that equals 2.

To simplify a radical expression, factor the number under the radical into its prime factors and look for pairs (for square roots) or groups (for cube roots) to simplify.

The product of two radicals can be simplified by multiplying the numbers under the radicals together. For example, $$\text{If } \text{A} = \text{√}a \text{ and } \text{B} = \text{√}b, \text{ then } \text{A} \times \text{B} = \text{√}(a \times b)$$ .

The sum of two like radicals is found by adding their coefficients. For example, $$3\text{√}2 + 4\text{√}2 = 7\text{√}2$$ .

The difference of two like radicals is found by subtracting their coefficients. For example, $$5\text{√}3 - 2\text{√}3 = 3\text{√}3$$ .

Use the distributive property (FOIL method) to multiply each term in the first binomial by each term in the second binomial. For example, $$(a + \text{√}b)(c + \text{√}d) = ac + a\text{√}d + c\text{√}b + \text{√}(bd)$$ .

The simplified form is $$\text{√}(a \times b)$$ .

The simplified form is $$a$$ , assuming a is non-negative.

$$3\text{√}9 = 3 \times 3 = 9, 7\text{√}25 = 7 \times 5 = 35, \text{√}64 = 8. So, 9 + 35 - 8 = 36$$ .

The result is $$\text{√}(3x^3 \times 12x) = \text{√}(36x^4) = 6x^2$$ .