if a=root 3 - root 2 ÷ root 3 + root 2 of b = root 3 + root 2 ÷ root 3 - root 2 find a square + b square = 5ab
a = (√3-√2)/(√3+√2)
b = (√3+√2)/(√3-√2)
Clearly ab=1
Doing the squares directly is a bit cumbersome, so let's rationalize the denominators:
a = (√3-√2)^2 = 5-2√6
b = (√3+√2)^2 = 5+2√6
Now we can take
a^2 = 49-20√6
b^2 = 49+20√6
a^2+b^2 = 98
Not sure that 98 = 5*1
To find the value of a^2 + b^2 in terms of a and b, we can follow these steps:
1. Simplify the expressions for a and b:
We have a = (sqrt(3) - sqrt(2)) / (sqrt(3) + sqrt(2)).
To simplify this, we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is (sqrt(3) - sqrt(2)).
So, a = ((sqrt(3) - sqrt(2)) * (sqrt(3) - sqrt(2))) / ((sqrt(3) + sqrt(2)) * (sqrt(3) - sqrt(2))).
Simplifying further, we get a = (3 - 2sqrt(6) + 2) / (3 - 2).
This becomes a = (5 - 2sqrt(6)) / 1.
Therefore, a = 5 - 2sqrt(6).
Similarly, we have b = (sqrt(3) + sqrt(2)) / (sqrt(3) - sqrt(2)).
Using the same rationalization process, we get b = (5 + 2sqrt(6)) / 1.
Therefore, b = 5 + 2sqrt(6).
2. Find the product of a and b:
We can find the product of a and b by multiplying the two expressions:
a * b = (5 - 2sqrt(6)) * (5 + 2sqrt(6)).
Applying the FOIL method, we get a * b = 25 - 4 * 6.
Simplifying further, we have a * b = 25 - 24 = 1.
3. Calculate 5 * a * b:
Multiplying the value obtained in step 2 by 5, we get 5 * a * b = 5 * 1 = 5.
4. Calculate a^2 + b^2:
To find a^2 + b^2, we need the square of a and b.
a^2 = (5 - 2sqrt(6))^2.
Expanding this expression, we get a^2 = 25 - 20sqrt(6) + 24.
Similarly, b^2 = (5 + 2sqrt(6))^2.
Expanding this expression, we get b^2 = 25 + 20sqrt(6) + 24.
Now, we can add a^2 and b^2: (25 - 20sqrt(6) + 24) + (25 + 20sqrt(6) + 24) = 98.
Therefore, a^2 + b^2 = 98.
Hence, the value of a^2 + b^2 is 98, which is not equal to 5ab.