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On this page, we hope to clear up problems that you might have with square roots and their uses.  Square roots are something you'll never get away from.  Reassuring, aren't we?  :-)    Scroll down or use the links below to start understanding square roots better! Simplification of square roots Addition and Subtraction of square roots Multiplication of square roots Quiz on Square Roots When simplifying square roots, keep the Product of Square Roots Theorem in mind.  It is outlined below: If m and n are not negative and are real numbers, then SQRT(m) * SQRT(n) = SQRT(mn). 1. Simplify: SQRT(50) Solution: Write 50 as a product of prime factors. SQRT(5 * 5 * 2) Use the Product of Square Roots Theorem to write the square root of above as a product of square roots. SQRT(5) * SQRT(5) * SQRT(2) By the definition of square roots, SQRT(5) * SQRT(5) = 5, so you now have 5(SQRT(2)). That is the answer! 2. Simplify: SQRT(147) Solution: Write 147 as a product of prime factors. SQRT(3 * 7 * 7) Use the Product of Square Roots Theorem to rewrite the root shown above. SQRT(3) * SQRT(7) * SQRT(7) Use the definition of square roots to simplify even further. The answer is 7(SQRT(3)). Back to Top  Adding and subtracting square roots is just like combining like terms when you need to do that with algebraic expressions.  If the indeces (a square root's index is 2, a cube root's index is 3, a 4th root's index is 4, etc.) or the radicands (the expression under the root sign or enclosed by parentheses after SQRT) are the same, you have a like term on your hands. 1. Add: (4 * SQRT(2)) - (5 * SQRT(2)) + (12 * SQRT(2)) Solution: Combine like terms by adding the numerical coefficients. (4 - 5 + 12) * SQRT(2) After the addition, you get the answer! 11 * SQRT(2) Many times, such problems will not be given to you with all the terms alike, or even trickier, the terms will only look different! 1. Simplify: SQRT(18) + SQRT(8) Solution: Write each square root as a product of prime factors. SQRT(2 * 3 * 3) + SQRT(2 * 2 * 2) Use the Product of Square Roots Theorem to rewrite each square root above. SQRT(2) * SQRT(3) * SQRT(3) + SQRT(2) * SQRT(2) * SQRT(2) Use the definition of square roots to simplify even further. (3(SQRT(2))) + (2(SQRT(2))) Add like terms to get an answer. 5(SQRT(2)) Back to top  Instead of using the Product of Square Roots Theorem to separate a root, we can use it in reverse to multiply radicals, as the following example shows:  SQRT(2) * SQRT(3) = SQRT(6) 1. Simplify: 4(SQRT(3)) * 3(SQRT(2)) Solution: As with anything else, changing the order of the factors does not change the problem, so we will rearrange the factors so the problem will be a little easier to compute. 4 * 3 * SQRT(3) * SQRT(2) Multiply all the factors together to get the answer. 12(SQRT(6)) Back to top  Take the Quiz on square roots.  (Very useful to review or to see if you've really got this topic down.)  Do it!