Question

Find the cube root of:
\[99 - 70\sqrt 2 \]

Answer 1

Hint: Try to convert the given number into an identity format, it will make the problem more simpler and since we are finding cube roots , we have to use a cubic identity.

Complete step-by-step answer:
We are given the number as
\[99 - 70\sqrt 2 \]
We have to split the given number such that either the identity ${(a - b)^3} = {a^3} - {b^3} + 3a{b^2} - 3{a^2}b$ or the identity ${(a + b)^3} = {a^3} + {b^3} + 3a{b^2} + 3{a^2}b$is followed.
Therefore, after splitting we get,
\[99 - 70\sqrt 2 \]\[ = 27 + 72 - 16\sqrt 2 - 54\sqrt 2 \]
Which can further be written as
\[ = {3^3} + {( - 2\sqrt 2 )^3} + 3 \times {3^2} \times ( - 2\sqrt 2 ) + 3 \times 3 \times {( - 2\sqrt 2 )^2}\]
It can be clearly seen that, it is of the form ${(a - b)^3} = {a^3} - {b^3} + 3a{b^2} - 3{a^2}b$
Where, $a = 3$ and $b = - 2\sqrt 2 $
Therefore, our value comes out to be
\[ = {(3 - 2\sqrt 2 )^3}\]
Therefore, we get
\[ \Rightarrow 99 - 70\sqrt 2 = {(3 - 2\sqrt 2 )^3}\]
Hence, the cube root of \[99 - 70\sqrt 2 \]is \[(3 - 2\sqrt 2 )\].

Note: In these types of questions, it is very important to split the given numbers in an appropriate formation of the identity, which can be further evaluated to reach an optimum solution.