Chapter 6.4, Problem 10WE

To determine

To Simplify:

  ${\left(3\sqrt{11}-\sqrt{10}\right)}^2$

Answer to Problem 10WE

  $109-6\sqrt{110}$

Explanation of Solution

Given:

The expression is :

  ${\left(3\sqrt{11}-\sqrt{10}\right)}^2$

Formula Used:

  ${\left(a-b\right)}^2=a^2-2ab+b^2$

Calculation:

Calculation:

Given:

  ${\left(3\sqrt{11}-\sqrt{10}\right)}^2\mathrm{..............}(1)$

Use

  ${\left(a-b\right)}^2=a^2-2ab+b^2\mathrm{..........}(2)$

Comparing equation (1) with (2),

  $a=3\sqrt{11},b=\sqrt{10}$

Substitute the values in equation (2),

  $\begin{array}{l}{\left(a-b\right)}^2\\ =a^2-2ab+b^2\\ ={\left(3\sqrt{11}\right)}^2-2\left(3\sqrt{11}\right)\left(\sqrt{10}\right)+{\left(\sqrt{10}\right)}^2\\ =3^2\cdot 11-6\sqrt{11\cdot 10}+\mathrm{10.................}[\text{Product property of radicals;}\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{ab}]\\ =99-6\sqrt{110}+10\\ =109-6\sqrt{110}\end{array}$