Question

`6^{(1)/(2) } +6^{(3)/(2) }`

Which of the following is equal to the value above?

`√(222)`

`7√(6)`

`6^{(3)/(4) }`

Answer 1

`7\sqrt6`

Explanation:

The original expression is
`6^{(1)/(2)} + 6^{(3)/(2)}`

The last one is a trick answer:
`6^{(1)/(2)} 6^{(3)/(2) }= 6^{(3)/(4)}` i.e. the exponents are added when the two are multiplied and the base of the exponent is the same(in this case 6)

`6^{(1)/(2) }= √(6) \\6^{(3)/(2) } = (6^{(1)/(2) })^3 = (√(6) )^3\\`

So
`6^{(1)/(2)} + 6^{(3)/(2)} = √(6) + (√(6) )^3`

Factoring out
`√(6)` in the above equation gives
`√(6) [1 + (√(6))^2]\\\\`

But
`(√(6) )^2 = 6` square of a square root of a number is the number

So the expression becomes
`√(6) (1 + 6) = 7\sqrt6`