To find the square root of an expression like $49 + 20 \sqrt{6}$ , we assume it can be written as the square of a binomial expression of the form $\sqrt{a} + \sqrt{b}$ or $a + \sqrt{b}$ . Let's assume:

$\sqrt{49 + 20 \sqrt{6}} = \sqrt{a} + \sqrt{b}$

Step 1: Square both sides to eliminate the square root:

$49 + 20 \sqrt{6} = (\sqrt{a} + \sqrt{b})^{2}$

Step 2: Expand the right side using the identity ${(x + y)}^{2} = x^{2} + y^{2} + 2 x y$ :

$49 + 20 \sqrt{6} = a + b + 2 \sqrt{a b}$

Step 3: Compare the rational and irrational parts on both sides. This gives us two equations:

1. $a + b = 49$ (from the rational part)

2. $2 \sqrt{a b} = 20 \sqrt{6}$ (from the irrational part)

Step 4: Simplify the second equation:

$\sqrt{a b} = 10 \sqrt{6}$

Square both sides: $a b = 100 \times 6 = 600$

Step 5: We now have:

$a + b = 49$ and $a b = 600$

We need to find two numbers whose sum is 49 and product is 600. Let's try factors of 600:

Possible pairs: (24,25), (20,30), (15,40), (12,50), (10,60), etc.

24 + 25 = 49 and 24 × 25 = 600. Perfect!

So, a = 24 and b = 25 (or vice versa).

Step 6: Substitute back into our assumed form:

$\sqrt{49 + 20 \sqrt{6}} = \sqrt{24} + \sqrt{25} = 2 \sqrt{6} + 5$

Therefore, the square root is $5 + 2 \sqrt{6}$ .

Final Answer: $5 + 2 \sqrt{6}$

Key Formula

The core method relies on the assumption and subsequent expansion:
$\sqrt{x + y \sqrt{z}} = \sqrt{a} + \sqrt{b}$
This leads to the system of equations:
$a + b = x$
$2 \sqrt{a b} = y \sqrt{z}$
which is solved to find the numbers a and b.

Related Topics

Key Formula