Question

Perform the following operations and write the answers and in radical form

Answer 1

The given expression is

`8^(1/2) +16^(1/4)-12^(1/2)+81^(1/4)`

We can write 8 as cube of 2, 16 ad fourth power of 2, 12 as square of 2 times 3 and 81 as fourth power of 3, that is

`(2^3)^(1/2) +(2^4)^(1/2) - (2^2 *3)^(1/2) + (3^4)^(1/2)`

In case of two powers, we have to multiply them.

`=2^(3/2) + 2^(4/2) -2^(4/2) * 3^(1/2) + 3^(4/2)`

Simplifying the exponent, will give

`=2*2^(1/2) +4 -4*3^(1/2) +9`

Combining like terms,

`=13+2*2^(1/2) -4*3^(1/2)`

Answer 2

`2√(2) +5-2√(3)`

Explanation:

Property of Radical expression is:

`(a^x)^{(y)/(z) } = a^{(xy)/(z) }`

Step to step solution of given expression is:

`8^{(1)/(2) }+16^{(1)/(4) }-12^{(1)/(2) }+81^{(1)/(4) }\\\Rightarrow (2^(3))^{(1)/(2) }+(2^(4))^{(1)/(4) }-(2^(2)*3)^{(1)/(2) }+(3^(4))^{(1)/(4) }\\\Rightarrow 2^{(3)/(2) } + 2 - 2√(3) +3\\\Rightarrow 2√(2) +5-2√(3)`