Question

If $A=(5\sqrt{2}+6\sqrt{3})(4\sqrt{2}-5\sqrt{3})$ and $B=(7\sqrt{2}+5\sqrt{3})(8\sqrt{2}-5\sqrt{3})$, then find the value of $A+B$

• A. $4\sqrt{6}-13$
• B. $7\sqrt{6}-13$
• C. $5\sqrt{6}-13$
• D. $3\sqrt{6}-13$

Answer 1

Answer: (A)

$4\sqrt{6}-13$

$A=(5\sqrt{2}+6\sqrt{3})(4\sqrt{2}-5\sqrt{3})$

$=\left(5\sqrt{2}\right)\left(4\sqrt{2}\right)-\left(5\sqrt{2}\right)\left(5\sqrt{3}\right)+\left(6\sqrt{3}\right)\left(4\sqrt{2}\right)-\left(6\sqrt{3}\right)\left(5\sqrt{3}\right)$

$=40-25\sqrt{6}+24\sqrt{6}-90$

$A=-50-\sqrt{6}$

$B=(7\sqrt{2}+5\sqrt{3})(8\sqrt{2}-5\sqrt{3})$

$=\left(7\sqrt{2}\right)\left(8\sqrt{2}\right)-\left(7\sqrt{2}\right)\left(5\sqrt{3}\right)+\left(5\sqrt{3}\right)\left(8\sqrt{2}\right)-\left(5\sqrt{3}\right)\left(5\sqrt{3}\right)$

$=112-35\sqrt{6}+40\sqrt{6}-75$

$B=37+5\sqrt{6}$

$A+B=-50-\sqrt{6}+37+5\sqrt{6}$

$=4\sqrt{6}-13$