Question

If AB√48,BC=2√30 and AC2√18 then ABC is

Answer 1

`12√(2)\sqrt[4]{5}`

or

`25.38`

Explanation:

We are given:

`AB = √( 48)\\\\BC = 2√(30)\\\\AC = 2√(18)\\\\`

Simplify each square root to the lowest radical:

`√( 48) = √(16 \cdot 3) = √(16)\cdot √(3) = 4 √(3)\\\\2√(30) = 2 √(3 \cdot 10) = 2 √(3) √(10)\\\\2√(18) = 2 √(9 \cdot 2 ) = 2 √(9) \cdot √(2) = 2 \cdot 3√(2) = 6√(2)\\\\`

Multiplying all three:

`4 √(3) * 2 √(3) √(10) * 3√(2)`

First multiply all terms not under square root:

4 x 2 x 6 = 48

Then multiply all the radicals which are the same:

`√(3) * √(3){√(10) * √(2)`

`√(10) = √(2)√(5)`

This works out to

`√(3){√(3)√(2)√(5)√(2)`

Rearranging the radicals

`√(3)√(3)√(2){√(2)√(5) = (√(3))^2 \cdot (√(2))^2 \cdot √(5)\\\\= 3 \cdot 2 \cdot √(5)\\\\= 6 √(5)\\\\`

Multiplying by 48, the value we got by multiplication of the terms not under square root we get this value as:

`48 * 6√(5)`

`= 288√(5)`

= 643.98757

This is equal to
`AB \cdot BC \cdot AC = A^2B^2C^2 = (ABC)^2`

`ABC = \sqrt{288√(5)} \\\\√(288) = √(144 \cdot 2) = 12 √(2)`

`\sqrt{√(5)} = \sqrt[4]{5}`

ABC =
`12√(2)\sqrt[4]{5}`

In decimal this would be

`√(643.98757) = 25.37691 \approx 25.38`