Question

9. Combine the radical expression, if possible.

Answer 1

we conclude that:

`3√(125)+4√(20)=23√(5)`

Hence, the last option i.e.
`23√(5)` is correct.

Explanation:

Given the expression

`3√(125)+4√(20)`

Combining the radical expressions

`3√(125)+4√(20)`

let us first solve

`3√(125)`

`=3√(5^3)`

`=3√(5^2\cdot \:5)`

Apply radical rule:
`√(ab)=√(a)√(b),\:\quad \:a\ge 0,\:b\ge 0`

`=3√(5^2)√(5)`

Apply radical rule:
`√(a^2)=a,\:\quad \:a\ge 0`

`=3\cdot \:5√(5)`

`=15√(5)`

Thus,

`3√(125)=15√(5)`

similarly solving

`4√(20)`

`=4√(2^2\cdot \:5)`

Apply radical rule:
`√(ab)=√(a)√(b),\:\quad \:a\ge 0,\:b\ge 0`

`=4√(2^2)√(5)`

Apply radical rule:
`√(a^2)=a,\:\quad \:a\ge 0`

`=4\cdot \:2√(5)`

`=8√(5)`

Thus,

`4√(20)=8√(5)`

so we get

`3√(125)=15√(5)`

`4√(20)=8√(5)`

so the expression becomes

`3√(125)+4√(20)=15√(5)+8√(5)` ∵
`3√(125)=15√(5)` ,
`4√(20)=8√(5)`

`=23√(5)` ∵
`15√(5)+8√(5)=23√(5)`

Therefore, we conclude that:

`3√(125)+4√(20)=23√(5)`

Hence, the last option i.e.
`23√(5)` is correct.