Simplify the expression inside the radical by performing any arithmetic that is called for.
For example, to simplify √( 2,000 + 200 - 40 ), add 2,000 and 200 together to yield 2,200, and then subtract 40 from this sum to yield 2,160. Thus, √( 2,000 + 200 - 40 ) simplifies to √(2,160); 2,160 is the simplified radicand.
Find the prime factorization of the radicand. The prime factorization of a number is the number expressed solely as a factor of primes. To find the prime factorization of a number, use a tool such as Wolfram Alpha for larger numbers. Navigate to the Wolfram Alpha website and enter "prime factorization of" followed by the simplified radicand.
The prime factorization of 2,160, for example, is 2^4 x 3^3 x 5. Prime factorization helps identify perfect squares.
Identify the prime factors that are perfect squares -- that is, the prime factors that are raised to a multiple of 2.
For the prime factorization of 2,160, 2^4 x 3^3 x 5, only one prime factor, 2, is raised to a multiple of two. In 2^4 x 3^3 x 5, 2 is raised to the power 4.
Divide by 2 the power on those factors raised to a multiple of 2 and move these factors outside the radical.
For example, to simplify √( 2^4 x 3^3 x 5 ), divide the 4 power of 2^4 by 2 to yield 2^2, since 4 divided by 2 is 2. Move the new 2^2 term outside the radical to obtain 2^2 √( 3^3 x 5 ). The terms √( 2^4 x 3^3 x 5 ) and 2^2 √( 3^3 x 5 ) are equivalent because √( 2^4 ) equals 2^2.
Identify the prime factors that are raised to a power such that subtracting 1 from the power makes it a perfect square -- that is, subtracting 1 from the power makes it a multiple of 2. Only identify such factors if they are raised to the power of 3 or greater.
For example, for the radicand 3^3 x 5, 3 is raised to the power 3, a number such that subtracting 1 from it would make it a multiple of 2, since 3 - 1 equals 2.
Subtract 1 from the power of the terms you just identified, divide the power on these terms by 2 and move the terms outside the radical. Keep one copy of the term, raised to the power 1, inside the radical.
For example, consider the case of 2^2 √( 3^3 x 5 ). Subtract 1 from the power of 3^3 to obtain 3^2. Divide the power on this term by 2 to yield 3^1, which is simply equivalent to 3. Move 3 outside the radical but keep one copy of 3, raised to the power 1, inside the radical, to obtain ( 2^2 x 3 ) √( 3 x 5 ). Note that any term moved outside the radical (3, in this case) is multiplied by any terms already outside the radical (2^2, in this case).
As a second example, consider √( 5^7 ). Subtract 1 from the power of 5^7 to obtain 5^6. Divide 6 by 2 to obtain 3 and move 5^3 outside the radical, keeping 5^1 inside the radical. The final simplification is thus 5^3 √( 5 ).
Multiply out all the terms inside the radical and outside the radical.
If your result is 3 √( 3 x 5 ), multiply out 3 x 5 to obtain a final simplification of 3√15. If your result is ( 2^2 x 3 )√( 3 x 5 ), multiply out 2^2 x 3 to obtain 12 and 3 x 5 to obtain 15. Substitute these numbers into ( 2^2 x 3 ) √( 3 x 5 ) to obtain 12√15.