The basic form looks like this:
a is $(x+y)$ and b is $(x·y)$
Let's go to the methods and examples.
PLUS
$$√{a + 2√b}$$
With $x > 0$ and $y > 0$
Example
$√{7 + 2√12}$
We have $x + y = 7$ and $xy = 12$
Factor pairs of $12$:
- $1 × 12$
- $2 × 6$
- $3 × 4$
We pick the last factor pair: $3 × 4$, because $3 + 4 = 7$
We have $x = 3$ and $y = 4$
Therefore, $√{7 + 2√12}$ can be simplified to $√3 + √4$ ➡️ $√3 + 2$
In conclusion:
MINUS
$$√{a - 2√b}$$
With $x > 0$, $y > 0$, and $x > y$
Example
$√{15 - 2√56}$
We have $x + y = 15$ and $xy = 56$
Factor pairs of $56$:
- $1 × 56$
- $2 × 28$
- $4 × 14$
- $7 × 8$
We pick the last factor pair: $7 × 8$, because $7 + 8 = 15$
But since $x > y$, then we put the $x = 8$ and $y = 7$
Therefore, $√{15 - 2√56}$ can be simplified to $√8 - √7$ ➡️ $2√2 - √7$
In conclusion:
Tricks: Radical Denesting Using Binomial Square Identity 🐴
That probably looks like mathematical jargon put together. But what it's trying to say is:
We're simplifying square root expressions using the identity:
From the basic identity:
You see, the form at the beginning of this article stems from this identity and we take the square root on both sides:
Final form:
⬆️ Similar to the beginning of this article.
Approved by Euler's ghost
Hm. Let's go then.
COMPRESSION
$√{12 + 4√8}$
We need to "compress" the $4$ to be $2$!
As such:
$√{12 + 4√8} = √{12 + 2·2√8} = √{12 + 2√{4·8}} = √{12 + 2√32}$
So now we have, $√{12 + 4√8} = √{12 + 2√32}$
Using the method from the examples above, we will get:
$√{12 + 2√32} = √8 + √4 = 2√2 + 2 = 2(√2 + 1)$ ✅
EXPANSION
$√{9 - √72}$
We need to summon the "2 coefficient" by extracting it from the $√72$! In other words, expand the 1 coefficient to 2.
Like so:
$√{9 - √72} = √{9 - √{4·18}} = √{9 - 2√18$
And then we have, $√{9 - √72} = √{9 - 2√18}$
Using the method from the examples above, we will get:
$√{9 - √72} = √{9 - 2√18} = √6 - √3$ ✅
🥳
