Simplifying Radicals?

Question

For some reason I just couldn't remember anything that my teacher explained today in class. So now I'm just looking at my homework and apparently I need to simplify these for a final answer, "-/(square root sign)119, -/80, and -/261" please help i haven't been able to find any good advice lately

Answer 1

Step 1: draw a factor tree. 

step 2: replace the number under the radical with the prime factorization of the number

 

step 3: Any Move all pairs out of the radical, place ONLY one copy of the 

          number outside the radical sign.

 

sqrt is the short hand (and computer function) for square root

EXAMPLE.   

  square root of 45 = sqrt(45)

 

step 1: prime factor --->  45 = 3 x 3 x 5

step 2: sqrt(45) = sqrt(3*3*5) 

step 3:  3 * sqrt(5)  <--- pair of 3's under the radical, moves it out.

 

The reason why you can move it out, is because the square root of a number squared is just that same number.

   Square roots and squares cancel each other out.

 

EX.  sqrt(80) = sqrt(10 x 8) = sqrt( 5 x 2 x 4 x 2) = sqrt( 5 x 2 x 2 x 2 x 2)

    There are two pairs of 2's under the radical , so they both come out.

     sqrt(80) = 2 * 2 * sqrt(5) = 4 * sqrt(5)

 

The beauty of this is that you can actually check your answer, by squaring it.

 You must come out with the same number you started.

 Ex. [ 3 x sqrt(5) ] ^2 = 3^2 x sqrt(5)^2  <---- property of exponents

                                 = 9  x 5 = 45 <--- yes, it checks.

 

 Ex. [ 4 * sqrt(5) ]^2 = 4^2 * sqrt(5)^2 = 16 * 5 = 80 <--- yes, it checks.

 

Now to your homework problems.

 

119  = 7 * 17.   It cannot be factored further, so there is no point 

  in rewriting anything. sqrt(119) is automatically simplified. Leaves it alone.

 

sqrt(80) has already been done for you above.

 

sqrt(261) = sqtt(3 * 87) = sqrt(3 * 3 * 29)

because the prime factorization of 261 = 3 x 3 x 29 ;

There is only one pair of 3's , so it simplifies to

3 * sqrt(29)

 

check: 3^2 * sqrt(29)^2 = 9 * 29 = 261.