Question

(12/√13-1 - 36/√13-1 + 2/1-√2) * (4+2√2)

Answer 1

Вначале упростим отдельно каждый член в сумме:

$\frac{12}{\sqrt{13}-1} =\frac{12*(\sqrt{13}+1)}{13-1} =\frac{12*(\sqrt{13}+1)}{12} =\sqrt{13}+1$

$\frac{36}{\sqrt{13}-1} =3*\frac{12}{\sqrt{13}-1} =3\sqrt{13}+3$

$\frac{2}{1-\sqrt{2}} =\frac{2*(1+\sqrt{2})}{1-2} =-2-2\sqrt{2}$

Теперь посчитаем левую скобку в сумме:

$\sqrt{13} +1-3\sqrt{13} -3-2-2\sqrt{2} =-4-2(\sqrt{13} +\sqrt{2} )$

Домножим левую скобку на правую:

$-(4+2\sqrt{2} )*(4+2\sqrt{2} )-2\sqrt{13} *(4+2\sqrt{2} )=-16-16\sqrt{2} -8-8\sqrt{13} -4\sqrt{26} =-4(6+4\sqrt{2} +2\sqrt{13} +\sqrt{26} )=-4((2+\sqrt{2})^2+\sqrt{13} *(2+\sqrt{2})=-4( (2+\sqrt{2})((2+\sqrt{2}+\sqrt{13} )$

Ответ: $-4( (2+\sqrt{2})((2+\sqrt{2}+\sqrt{13} )$

Answer 2

Let's simplify the expression step by step:

1. First, let's simplify the expressions within the parentheses:

   • 12/√13-1 = 12/(√13 - 1)
   • 36/√13-1 = 36/(√13 - 1)
   • 2/1-√2 = 2/(1 - √2)

2. Now, let's find a common denominator for these three fractions, which is (√13 - 1)(1 - √2):

   • 12/(√13 - 1) * ((1 + √2)/(1 + √2)) = (12(1 + √2))/((√13 - 1)(1 + √2))
   • 36/(√13 - 1) * ((1 + √2)/(1 + √2)) = (36(1 + √2))/((√13 - 1)(1 + √2))
   • 2/(1 - √2) * ((√13 + 1)/(√13 + 1)) = (2(√13 + 1))/((1 - √2)(√13 + 1))

3. Now, we have:

   [(12(1 + √2))/((√13 - 1)(1 + √2)) - (36(1 + √2))/((√13 - 1)(1 + √2)) + (2(√13 + 1))/((1 - √2)(√13 + 1))] * (4 + 2√2)

4. Combine the numerators over the common denominator:

   [((12(1 + √2)) - (36(1 + √2)) + (2(√13 + 1))) / ((√13 - 1)(1 + √2))] * (4 + 2√2)

5. Now, let's simplify the numerator:

   (12(1 + √2) - 36(1 + √2) + 2(√13 + 1)) = 12(1 + √2 - 3(1 + √2) + 2(√13 + 1))

   = 12(1 - 3 + √2 - 3√2 + 2√13 + 2)

   = 12(-2 - 2√2 + 2√13)

   = -24 - 24√2 + 24√13

6. Now, we have:

   [-24 - 24√2 + 24√13] / ((√13 - 1)(1 + √2)) * (4 + 2√2)

7. Distribute the (4 + 2√2) to the numerator:

   [-24(4 + 2√2) - 24√2(4 + 2√2) + 24√13(4 + 2√2)] / ((√13 - 1)(1 + √2))

8. Now, simplify each term in the numerator:

   -96 - 48√2 - 96√2 - 48√8 + 96√13 + 48√26 / ((√13 - 1)(1 + √2))

9. Combine like terms:

   (-96 - 96√2 - 48√2 - 48√8 + 96√13 + 48√26) / ((√13 - 1)(1 + √2))

   = (-144 - 144√2 - 48√8 + 96√13 + 48√26) / ((√13 - 1)(1 + √2))

10. Finally, if you want to rationalize the denominator, you can multiply both the numerator and denominator by the conjugate of the denominator:

= [(-144 - 144√2 - 48√8 + 96√13 + 48√26) * (√13 + 1)] / [((√13 - 1)(1 + √2)) * (√13 + 1)]

= [(-144√13 - 144 - 144√26 - 48√8√13 - 48√8 + 96√13√13 + 96√13 + 48√26√13 + 48√26)] / [(√13 - 1)(1 + √2)(√13 + 1)]

= [(-144√13 - 144 - 144√26 - 48√104 - 48√8 + 96*13 + 96√13 + 48√338 + 48√26)] / [(√13 - 1)(1 + √2)(√13 + 1)]

Now, you have a simplified expression. You can further simplify or approximate the numerical value if needed.