Question

What is the simplest radical form for $\sqrt{145}$?

Answer 1

Hint: For the given number 145 we will write it as a product of its prime factors. Now, if there will be the same prime factors more than once then we will write it in the exponential form and use the property of exponents ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to simplify the radical expression. If all the prime factors will be different then we will not be able to simplify $\sqrt{145}$ any further and it will be considered as the simplest form.

Complete step by step answer:
Here we have been provided with the expression $\sqrt{145}$ and we are asked to write its simplest radical form. Let us assume this expression as E. So, we have,
$\Rightarrow E=\sqrt{145}$
Now, to simplify the above expression further we need to write it as the product of its prime factors. Since, we have the under root sign in the radical expression so we will try to form pairs of two similar prime factors if they will appear and we will use the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to remove them from the radical sign. The factors which will appear only once will be left inside the radical sign. In case all the prime factors are different then the provided expression cannot be simplified further. So we can write,
\[\Rightarrow 145=5\times 29\]
Clearly we can see that there are two different prime factors so we cannot take out any factor out of the radical sign, therefore $\sqrt{145}$ cannot be simplified further.
$\therefore E=\sqrt{145}$

Note: You may note that here you do not have to find the square root but only simplify it so do not try to use the long division method to get the answer in decimal form. Remember all the formulas of exponents in case they can be used in other expressions.