In general, for positive values of x,

Let's try reversing the order of the previous examples?


For example, 4 can be expressed in entire radical form, as follows.

Example 4

These relationships can be applied when converting mixed radicals to entire radicals.

For example, = .

Example 5

Express as an entire radical.

Step 1
Rewrite as an entire radical by squaring the 3
(taking it to the power of 2).

Step 2
Bring the and the 5 under the same radical sign and express the result as a product.

Step 3
Multiply the factors.


Many professionals, including physicists and engineers, work with radicals to solve complicated problems involving space travel and construction. Square roots, cube roots, fourth roots, and beyond are not uncommon in these applications. Expressing solutions in exact value form, such as entire and simplified mixed radicals, help diminish the complexity of certain solutions. As with estimating grocery and restaurant bills, it is always important to have a sense of what a reasonable answer to a problem involving radicals might look like. This estimation can be the difference between a building standing tall and a building that collapses.