MIT grad shows how to simplify radical expressions, specifically square root expressions, into their simplest form ("Simplified Radical Form" or "SRF Form"). To skip ahead: 1) for a PERFECT SQUARE under the root like sqrt(16) skip to time 1:29. 2) for a SMALL number under the root that is NOT a perfect square like sqrt(32), skip to time 2:45. 3) for a LARGE number under the root, such as sqrt(343), skip to 5:14. 4) for a PRODUCT, meaning a number times a square root like 5*sqrt(54) skip to 7:30. 5) for a QUOTIENT, meaning a number divided by a square root like 4/sqrt(2), and where you have to RATIONALIZE the denominator, skip to 10:10. and 6) for a more complex QUOTIENT example like (4+sqrt(3))/(5-sqrt(3)) where you have to multiply by the CONJUGATE to simplify, skip to 12:36. Nancy formerly of MathBFF explains the steps.
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For all simplifying radical problems:
1) First try to find a perfect square number that divides evenly into the number under the square root. Perfect square numbers include 4, 9, 16, 25, 36, 49, etc. because they are squares of the numbers 2, 3, 4, 5, 6, 7, etc., respectively.
2) Separate the root expression into the two numbers you found as factors, with each number now under its own square root symbol. For example, sqrt(12) = sqrt(4) times sqrt(3).
3) Keep simplifying. The square root of a perfect square becomes just a number, without the radical square root symbol. sqrt(4)*sqrt(3) becomes 2*sqrt(3).
4) Check to see if anymore simplifying can be done, either by combining constants or finding another perfect square number that divides evenly into your factors. If no more simplifying can be done, you have your final "simplified radical form" (SRF) answer, 2*sqrt(3).
If there is a square root anywhere in the DENOMINATOR, you must "rationalize" the denominator, in other words, somehow get rid of the root in the bottom, in order for the expression to be considered simplest. For example, if you have 4/sqrt(2) you must use the trick of simplifying by multiplying the expression by sqrt(2)/sqrt(2). This will clear the square root in the bottom. If you multiply straight across, you would get 4*sqrt(2)/2, which simplifies again to 2*sqrt(2).
If you have a more complicated fraction with a square root in the denominator, like (4+sqrt(3))/(5-sqrt(3)), you will need to use the "conjugate" to simplify. This means multiplying top and bottom by the expression in the denominator, but with the middle sign flipped. Here you would multiply by (5+sqrt(3))/(5+sqrt(3)), multiply straight across on top and bottom using FOIL-ing and distribution. Some terms will cancel in the bottom, leaving you with a simpler denominator that has no square roots.
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