Question 276871
Simplifying square roots is a matter of finding factors that are perfect squares.<br>
{{{sqrt(7825)}}}
Are there perfect square factors in 7825? One that should be obvious is 25. Factoring out 25 we get:
{{{sqrt(25*313)}}}
Using the property of radicals, {{{root(a, p*q) = root(a, p) * root(a, q)}}} we can separate the two factors into their own square roots:
{{{sqrt(25)*sqrt(313)}}}
And we can replace the first square root with its value of 5:
{{{5sqrt(313)}}}
(You can check but I don't think 313 has any perfect square factors. If this is true then twe are done.<br>
{{{sqrt(16524)}}}
Again we look for perfect square factors. Since the last two digits are divisible by 4 then the whole number is divisible by 4 (and 4 is a perfect square).
{{{sqrt(4*4131)}}}
Since the digits of 4131 add up to a number which is divisible by 9, then the entire number is divisible by 9 (which is a perfect square):
{{{sqrt(4*9*459)}}}
The digits of 459 also add up to a number divisible by 9:
{{{sqrt(4*9*9*51)}}}
51 has not perfect square factors so we are finished with the factoring. Now we split up the square root:
{{{sqrt(4)*sqrt(9)*sqrt(9)*sqrt(51)}}}
{{{2*3*3*sqrt(51)}}}
{{{18sqrt(51)}}}<br>
I'll leave the last one for you.