Equivalent Radical

Two radical expressions are equivalent radical expressions if they represent the same real number.

A Practical Example

The square root of 9 and the cube root of 27 are equivalent radical expressions, since both evaluate to 3.

$$ \sqrt{9} = 3 $$

$$ \sqrt[3]{27} = 3 $$

A further example is provided by the square root of 4 and the cube root of 8.

$$ \sqrt{4} = 2 $$

$$ \sqrt[3]{8} = 2 $$

Here again, the expressions yield the same numerical value.

Note. In general, an equivalent radical expression may be obtained by multiplying both the index of the root and the exponent of the radicand by the same natural number p, with p greater than zero. $$ \sqrt[n]{a^m} = \sqrt[n \cdot p]{a^{m \cdot p}} $$ The resulting expression is equivalent to the original. An equivalent form is also obtained by dividing both the index and the exponent by a common divisor. $$ \sqrt[n]{a^m} = \sqrt[\frac{n}{p}]{a^{\frac{m}{p}}} $$ Consequently, every radical expression admits infinitely many equivalent representations. This principle underlies the invariant property of radicals.

And so on.