Simplify √6(√54 − √(1/6)) + √((−7)²) − (−2√6)².

Step-by-step solution

1. 1

   Simplify the surds inside the bracket.

   $\sqrt{54}=\sqrt{9\cdot 6}=3\sqrt6,\qquad \sqrt{\tfrac16}=\frac{1}{\sqrt6}=\frac{\sqrt6}{6}$

   The second one is rationalised by multiplying numerator and denominator by $\sqrt6$.

2. 2

   Combine inside the bracket.

   $3\sqrt6-\frac{\sqrt6}{6}=\frac{18\sqrt6-\sqrt6}{6}=\frac{17\sqrt6}{6}$

3. 3

   Multiply by the outside √6. Since $\sqrt6\cdot\sqrt6=6$,

   $\sqrt6\cdot\frac{17\sqrt6}{6}=\frac{17\cdot 6}{6}=17$

4. 4

   Evaluate the two remaining terms carefully. These are the two traps:

   $\sqrt{(-7)^{2}}=\sqrt{49}=7\quad(\text{not }-7,\text{ since }\sqrt{a^{2}}=|a|)$

   $\left(-2\sqrt6\right)^{2}=(-2)^{2}\left(\sqrt6\right)^{2}=4\cdot 6=24\quad(\text{positive, because it is a square})$

5. 5

   Add everything up.

   $17+7-24=0$

   $\boxed{0}$

6. 6

   Verify numerically. $\sqrt6\approx 2.4495$, $\sqrt{54}\approx 7.3485$, $\sqrt{1/6}\approx 0.4082$. Then $2.4495(7.3485-0.4082)=2.4495(6.9402)=17.000$, and $17+7-24=0$ ✓.