Fill in the blanks so as to make each of the following statements true (Final answer only)

$\sqrt{1 \frac{25}{144}}=\ldots \ldots$

Insert a rational number and an irrational number between the following:

$2.357$ and $3.121$

Find the value of $a$ :

$\frac{3-\sqrt{5}}{3+2 \sqrt{5}}=a \sqrt{5}-\frac{19}{11}$

prove that.

$\left(1^{3}+2^{3}+3^{3}+4^{3}+5^{3}\right)^{\frac{1}{2}}$ $=\left(1^{3}+2^{3}+3^{3}+4^{3}\right)^{\frac{1}{2}}+\left(5^{3}\right)^{\frac{1}{3}}$

Find the value

$(343)^{-\frac{2}{3}}$

Rationalise the denominator in each of the following

$\frac{4}{\sqrt{10}+\sqrt{6}}$