6√50 = 6√(25 x 2) = (6 x 5)√2 = 30√2. Here, you’ve factored “50” into “25 x 2” and then have pulled out the “5” from the perfect square, “25”, and placed it outside of the radical, with the “2” remaining on the inside. Then, you multiplied “5” by “6”, the number already outside the radical, to get 30 as the new coefficient. 2√8 = 2√(4 x 2) = (2 x 2)√2 = 4√2. Here, you’ve factored “8” into “4 x 2” and then have pulled out the “2” from the perfect square “4” and placed it outside the radical, leaving the “2” on the inside. Then, you multiplied “2” by “2”, the number already outside the radical, to get 4 as the new coefficient. 5√12 = 5√(4 x 3) = (5 x 2)√3 = 10√3. Here, you’ve factored “12” into “4 x 3” and have pulled out the “2” from the perfect square “4” and placed it outside the radical, leaving the factor “3” on the inside. Then, you multiplied “2” by “5”, the number already outside the radical, to get 10 as the new coefficient.
30√2 - 4√2 + 10√3 = (30 - 4)√2 + 10√3 = 26√2 + 10√3
Simplify √(45). First, you can factor it out to get √(9 x 5). Then, you can pull out a “3” from the perfect square, “9,” and make it the coefficient of the radical. So, √(45) = 3√5. [6] X Research source Now, just add up the coefficients of the two terms with matching radicands to get your answer. 3√5 + 4√5 = 7√5
Simplify 6√(40). First you can factor out “40” to get “4 x 10”, which makes 6√(40) = 6√(4 x 10). Then, you can pull out a “2” from the perfect square, “4,” and then multiply it by the current coefficient. Now you’ve got 6√(4 x 10) = (6 x 2)√10. Multiply the two coefficients to get 12√10. Now, your problem reads 12√10 - 3√(10) + √5. Since the first two terms have the same radicand, you can subtract the second term from the first and leave the third as it is. You’re left with (12-3)√10 + √5, which can be simplified to 9√10 + √5.
Since √9 is equal to √(3 x 3), you can simplify √9 to 3. Since √4 is equal to √(2 x 2), you can simplify √4 to 2. Now, you can simply add 3 + 2 to get 5. Since 5 and 3√2 are not like terms, there’s nothing more you can do. Your final answer is 5 - 3√2.
Make it so these terms have the same denominator. The lowest common denominator, or the denominator that would be evenly divisible by both the denominators “4” and “2,” is “4. “[7] X Research source So, to make the second term, (√2)/2, have the denominator of 4, you need to multiply both its numerator and denominator by 2/2. (√2)/2 x 2/2 = (2√2)/4. Add up the numerators of the fractions while leaving the denominator the same. Do just what you would do if you were adding fractions. (√2)/4 + (2√2)/4 = 3√2)/4.
Note: saying the “half power of (2x)” = (2x)1/2 is just another way to say “square root of (2x)”.
