Question

Find the square root: 30+12√6

Answer 1

Answer:

3√2 - 2√3 or 0.77854

Step-by-step explanation:

To find the square root of the binomial quadratic surd 30–12√6, we asume

√(30–12√6) = √x - √y ……………………………………………………………………(1)

then‡ √(30+12√6) = √x + √y ..……………………………………………………. …(2)

Since the law of indices holds good for surds, multiplication of (1) and (2) gives

√[(30–12√6)(30+12√6)] = (√x - √y)(√x + √y)

Or, √[30² - (12√6)²] = (√x)² - (√y)² = x-y

Or, x - y = √(900 - 144x6) =√(900 - 864) = √36 = ±6

Since every square root has the double sign, every quadratic surd has two values, one positive and one negative. But we shall consider only the positive root as nothing to the contrary is expressly stated, . Thus

x - y = 6………………………………………………………………………………………(3)

Squaring (1),

30–12√6 = (√x - √y)² = x + y - 2√(xy)

Equating the rational parts

x + y = 30……………………………………………………………………………………(4)

whence , from (3) and (4)

x = 18, y = 12 . Substituting in (1),

√(30–12√6)=√18-√12 = 3√2 - 2√3 = 3x1.41421—2x1.73205=4.24264–3.46410= 0.77854

Hence, √(30–12√6) = 3√2 - 2√3 or 0.77854 (Proved)

Verification:

R.H.S. = 3√2 - 2√3 , Squaring

(3√2 - 2√3)² = 18 + 12 -12√6 = 30 - 12√6 = L.H.S.

‡ If √(a-√b) =√x - √y, then by squaring

a - √b = x + y - 2√xy

Equating the rational and the irrational parts,

a = x+y, √b = 2√xy

Hence, a + √b= x+y+2√xy = (√x+√y)²

that is, √(a + √b)= √x + √y

I HOPE IT HELPS

Answer 2

Answer

Step-by-step explanation: