Hint: First see whether the product of given numbers is a perfect square or not, if not then use the formula $\sqrt{ab}$ to introduce the irrational number between given two numbers. Then similarly find the irrational number between one of the given numbers and the newly introduced irrational number.
Complete step-by-step answer:
To solve the given question we will write the given data first,
Given two numbers as, 0.7 and 0.77 …………………………………………………. (1)
Now to introduce two irrational numbers between given numbers we will introduce one first and for that we should know the formula given below,
Formula: If ‘a’ and ‘b’ be two numbers such that their multiplication i.e. ‘ab’ is not a perfect square then the irrational number between ‘a’ and ‘b’ is given by,
Irrational number = $\sqrt{ab}$
As we have given the two numbers 0.7 and 0.77 and their multiplication is given by, $0.7\times 0.77=0.539$ is not a perfect square therefore by using the above formula the irrational number between 0.7 and 0.77 is given by,
Irrational number = $\sqrt{0.7\times 0.77}$
Therefore irrational number = $\sqrt{0.539}$ ………………………………………………… (2)
Now we will introduce a irrational number between 0.7 and $\sqrt{0.539}$ which is ultimately in between 0.7 and 0.77 as follows,
As we have given the two numbers 0.7 and $\sqrt{0.539}$ and their multiplication is given by, $0.7\times \sqrt{0.539}$ is not a perfect square therefore by using the above formula the irrational number between 0.7 and $\sqrt{0.539}$ is given by,
Irrational number = $\sqrt{0.7\times \sqrt{0.539}}$
Therefore irrational number = $\sqrt{0.5115}$ ………………………………………………… (3)
From equation (2) and equation (3) we can write the answer as $\sqrt{0.539}$ and $\sqrt{0.5115}$ are two irrational numbers between 0.7 and 0.77.
Note: There is no necessity to introduce an irrational number between 0.7 and $\sqrt{0.539}$ you can introduce it between $\sqrt{0.539}$ and 0.77 also which can also be the irrational number between 0.7 and 0.77.