Number Systems and Radical Operations

A radical is an expression that includes a root, such as a square root (√) or cube root (∛). For example, $$\text{√}x$$ represents the square root of x.

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

Like radicals are radicals that have the same index and the same radicand. To add like radicals, you add their coefficients while keeping the radical part unchanged.

To rationalize the denominator, multiply the numerator and denominator by a suitable radical that will eliminate the radical in the denominator.

The rationalized form is $$\frac{7\text{√}5}{5}$$ .

$$2\text{√}24 = 2\text{√}(4 \times 6) = 2 \times 2\text{√}6 = 4\text{√}6$$ .

$$\frac{1}{2}\text{√}80 = \frac{1}{2}\text{√}(16 \times 5) = \frac{1}{2} \times 4\text{√}5 = 2\text{√}5$$ .

Like radicals are radicals that have the same index and radicand. For example, $$\text{√}5$$ and $$3\text{√}5$$ are like radicals.

The sum is $$(5 + 2)\text{√}3 = 7\text{√}3$$ .

Unlike radicals have different radicands or different indices. For example, $$\text{√}2$$ and $$\text{√}3$$ are unlike radicals.