information:filtering
Simplified form of a radical expression

A non-nested radical expression is said to be in simplified form if no factor of the radicand can be written as a power greater than or equal to the index; there are no fractions inside the radical sign; and there are no radicals in the denominator.

For example, to write the radical expression 32 / 5 {\displaystyle \textstyle {\sqrt {32/5}}} in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:

32 5 = 16 ⋅ 2 5 = 16 ⋅ 2 5 = 4 2 5 {\displaystyle {\sqrt {\frac {32}{5}}}={\sqrt {\frac {16\cdot 2}{5}}}={\sqrt {16}}\cdot {\sqrt {\frac {2}{5}}}=4{\sqrt {\frac {2}{5}}}}

Next, there is a fraction under the radical sign, which we change as follows:

4 2 5 = 4 2 5 {\displaystyle 4{\sqrt {\frac {2}{5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}}

Finally, we remove the radical from the denominator as follows:

4 2 5 = 4 2 5 ⋅ 5 5 = 4 10 5 = 4 5 10 {\displaystyle {\frac {4{\sqrt {2}}}{\sqrt {5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}\cdot {\frac {\sqrt {5}}{\sqrt {5}}}={\frac {4{\sqrt {10}}}{5}}={\frac {4}{5}}{\sqrt {10}}}

When there is a denominator involving surds it is always possible to find a factor to multiply both numerator and denominator by to simplify the expression. For instance using the factorization of the sum of two cubes:

1 a 3 + b 3 = a 2 3 − a b 3 + b 2 3 (a 3 + b 3) (a 2 3 − a b 3 + b 2 3) = a 2 3 − a b 3 + b 2 3 a + b. {\displaystyle {\frac {1}{{\sqrt{a}}+{\sqrt{b}}}}={\frac {{\sqrt{a^{2}}}-{\sqrt{ab}}+{\sqrt{b^{2}}}}{\left({\sqrt{a}}+{\sqrt{b}}\right)\left({\sqrt{a^{2}}}-{\sqrt{ab}}+{\sqrt{b^{2}}}\right)}}={\frac {{\sqrt{a^{2}}}-{\sqrt{ab}}+{\sqrt{b^{2}}}}{a+b}}.}

Simplifying radical expressions involving nested radicals can be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advanced Galois theory. Moreover, when complete denesting is impossible, there is no general canonical form such that the equality of two numbers can be tested by simply looking at their canonical expressions.

For example, it is not obvious that

3 + 2 2 = 1 + 2. {\displaystyle {\sqrt {3+2{\sqrt {2}}}}=1+{\sqrt {2}}.}

The above can be derived through:

3 + 2 2 = 1 + 2 2 + 2 = 1 2 + 2 2 + 2 2 = (1 + 2) 2 = 1 + 2 {\displaystyle {\sqrt {3+2{\sqrt {2}}}}={\sqrt {1+2{\sqrt {2}}+2}}={\sqrt {1^{2}+2{\sqrt {2}}+{\sqrt {2}}^{2}}}={\sqrt {\left(1+{\sqrt {2}}\right)^{2}}}=1+{\sqrt {2}}}

Let r = p / q {\displaystyle r=p/q}, with p and q coprime and positive integers. Then r n = p n / q n {\displaystyle {\sqrt{r}}={\sqrt{p}}/{\sqrt{q}}} is rational if and only if both p n {\displaystyle {\sqrt{p}}} and q n {\displaystyle {\sqrt{q}}} are integers, which means that both p and q are n th powers of some integer.