Question

What is the value of (a-b)3 when ab=36 and a3-b3=665

Answer 1

Answer:

Given : a^3 + b^3 = 35a3+b3=35 and ab = 6ab=6

Let us take the value of (a+b)=x(a+b)=x

We know that the value of  

( a + b ) ^ { 3 } = a ^ { 3 } + b ^ { 3 } + 3 a b ( a + b )(a+b)3=a3+b3+3ab(a+b)

Substituting (a + b) = x in the above expression, we get,

( x ) ^ { 3 } = a ^ { 3 } + b ^ { 3 } + 3 a b ( x )(x)3=a3+b3+3ab(x)

Substituting the value of ab = 6ab=6 and a^3 + b^3 = 35a3+b3=35 in the above derived expression

\begin{gathered}\begin{array} { c } { x ^ { 3 } = 35 + 3 \times 6 ( x ) } \\\\ { x ^ { 3 } = 35 + 18 x } \\\\ { x ^ { 3 } - 18 x = 35 } \\\\ { x \left( x ^ { 2 } - 18 \right) = 35 } \\\\ { x \left( x ^ { 2 } - 18 \right) = 5 \times 7 } \end{array}\end{gathered}x3=35+3×6(x)x3=35+18xx3−18x=35x(x2−18)=35x(x2−18)=5×7

Let us take

x=5x=5

And

\begin{gathered}\begin{array} { c } { x ^ { 2 } - 18 = 7 } \\\\ { x ^ { 2 } = 7 + 18 } \\\\ { x ^ { 2 } = 25 } \\\\ { x = 5 } \end{array}\end{gathered}x2−18=7x2=7+18x2=25x=5

Thus, the value of x is 5.

x=a+b=5x=a+b=5

Thus, the value of a + b is equal to 5