How do I transform this into simple radical form... I have no idea how to do this could you please tell me the logic behind how you do this and step by step thanks
(5^(1/2)-3^(1/2))^2
thanks again
Also can you tell me how to transform this one into radical form also
18^(1/2)-3(50)^(1/2)+5(80)^(1/2)-2(125)^(1/2)
on the first, FOIL it.
on the second
18=9*2
50=25*2
80=16*5
125=25*5
Notice I factored out the squares, this will leave you with two terms, sqrt2, and sqrt5 terms.
To transform an expression into simple radical form, you need to simplify any radicals present. Here's how to do it step by step for your first expression:
Expression: (5^(1/2) - 3^(1/2))^2
Step 1: Start by simplifying the square root terms individually.
√5 can be simplified as 2√5 because 5 is not a perfect square.
Similarly, √3 cannot be simplified further.
So the expression becomes: (2√5 - √3)^2
Step 2: Apply the square of a binomial formula. (a - b)^2 = a^2 - 2ab + b^2.
(2√5 - √3)^2 = (2√5)^2 - 2(2√5)(√3) + (√3)^2
= 4(5) - 2(2)(√15) + 3
= 20 - 4√15 + 3
= 23 - 4√15
Therefore, the simple radical form of the expression (5^(1/2) - 3^(1/2))^2 is 23 - 4√15.
Now let's move to your second expression:
Expression: 18^(1/2) - 3(50)^(1/2) + 5(80)^(1/2) - 2(125)^(1/2)
Step 1: Simplify the square root terms individually.
√18 = 3√2 (since 18 = 9 * 2, and 9 is a perfect square.)
√50 = 5√2 (since 50 = 25 * 2)
√80 = 4√5 (since 80 = 16 * 5)
√125 = 5√5 (since 125 = 25 * 5)
So the expression becomes: 3√2 - 3(5√2) + 5(4√5) - 2(5√5)
Step 2: Simplify accordingly.
3√2 - 3(5√2) + 5(4√5) - 2(5√5) = 3√2 - 15√2 + 20√5 - 10√5
Combine like terms:
(3√2 - 15√2) + (20√5 - 10√5) = -12√2 + 10√5
Therefore, the simple radical form of the expression 18^(1/2) - 3(50)^(1/2) + 5(80)^(1/2) - 2(125)^(1/2) is -12√2 + 10√5.