How do you evaluate 4√112 +5√56 - 9√126?

Jul 9, 2018

$16 \sqrt{7} - 17 \sqrt{14}$

Explanation:

Note that

$112 = 7 \cdot 16$

$56 = 4 \cdot 14$

$126 = 9 \cdot 14$

and

$\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$ if $a , b \ge 0$
so we get

$16 \sqrt{17} + 10 \sqrt{14} - 27 \sqrt{14}$

combining like Terms

$16 \sqrt{7} - 17 \sqrt{14}$

Jul 9, 2018

$16 \sqrt{7} - 17 \sqrt{14}$

Explanation:

Write each radicand (the number under the root) as a product of its factors. Try to use perfect squares wherever possible.

$4 \sqrt{112} + 5 \sqrt{56} - 9 \sqrt{126}$

$= 4 \sqrt{16 \times 7} + 5 \sqrt{4 \times 2 \times 7} - 9 \sqrt{9 \times 14}$

Find any roots possible:

$= 4 \times 4 \times \sqrt{7} + 5 \times 2 \sqrt{2 \times 7} - 9 \times 3 \sqrt{2 \times 7}$

$= 16 \sqrt{7} + 10 \sqrt{14} - 27 \sqrt{14}$

$= 16 \sqrt{7} - 17 \sqrt{14} \text{ } \leftarrow$ this can be the answer

Or, we can try to improve it a bit

$= 16 \sqrt{7} - 17 \sqrt{7} \sqrt{2}$

This can be factored to give:

$\sqrt{7} \left(16 - 17 \sqrt{2}\right)$

However, this answer is no better than the first.

Jul 9, 2018

$16 \sqrt{7} - 17 \sqrt{14}$

Explanation:

We have the following:

$4 \sqrt{112} + 5 \sqrt{56} - 9 \sqrt{126}$

In each of our terms, we can rewrite the radicals in such a way where we can factor out a perfect square:

$\textcolor{b l u e}{4 \sqrt{16 \cdot 7}} + \textcolor{p u r p \le}{5 \sqrt{4 \cdot 14}} - \textcolor{s t e e l b l u e}{9 \sqrt{9 \cdot 14}}$

$\textcolor{b l u e}{16 \sqrt{7}} + \textcolor{p u r p \le}{10 \sqrt{14}} - \textcolor{s t e e l b l u e}{27 \sqrt{14}}$

Since the latter two terms have a $\sqrt{14}$ in common, we can simplify those to get

$\textcolor{b l u e}{16 \sqrt{7}} - \textcolor{red}{17 \sqrt{14}}$

Since the radicals have no perfect square factors, we are done!

Hope this helps!

Jul 9, 2018

$16 \sqrt{7} - 17 \sqrt{14}$

Explanation:

$4 \sqrt{112} + 5 \sqrt{56} - 9 \sqrt{126}$

$\therefore = 4 \sqrt{7} \cdot \sqrt{16} + 5 \sqrt{7} \cdot \sqrt{8} - 9 \sqrt{2 \cdot 3 \cdot 3 \cdot 7}$

$\therefore = 4 \cdot 4 \sqrt{7} + 5 \cdot \sqrt{7} \cdot \sqrt{8} - 27 \cdot \sqrt{2} \cdot \sqrt{7}$

$\therefore = \sqrt{7} \left(16 + 5 \sqrt{2 \cdot 2 \cdot 2} - 27 \sqrt{2}\right)$

$\therefore = \sqrt{7} \left(16 + 10 \sqrt{2} - 27 \sqrt{2}\right)$

$\therefore = \sqrt{7} \left(16 + \sqrt{2} \left(10 - 27\right)\right)$

$\therefore = \sqrt{7} \left(16 + \sqrt{2} \left(- 17\right)\right)$

$\therefore = \sqrt{7} \left(16 - 17 \sqrt{2}\right)$

$\therefore = 16 \cdot \sqrt{7} - 17 \sqrt{2} \cdot \sqrt{7}$

$\therefore = 16 \sqrt{7} - 17 \sqrt{14}$