If a - b = 6 and ab = 20, find the value of a^3 - b^3. - Vidyadip

Question

If $a - b = 6$ and $ab = 20,$ find the value of $a^3 - b^3.$

✓

Answer

We have,
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
$= (a - b)(a^2 + ab + b^{2 }- 2ab + 2ab) [$Adding and substracting $2ab$ in the second break$]$
$= (a - b)\big[(a^2 + b^2 - 2ab) + 3ab\big]$
$= (a - b)\big[(a - b)^2 + 3ab\big] \big [\because (a - b)^2 = a^2 + b^2 - 2ab\big]$
$= 6 \times \big[(6)^2 + 3 \times 20\big] \big[\because a - b = 6$ and $ab = 20\big]$
$= 6 \times [36 + 60]$
$= 6 \times 96$
$= 576$
$\therefore a^3 - b^3 $
$= 576$

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