Question

9. Combine the radical expression, if possible.

Answer 1

Answer:

we conclude that:

$3\sqrt{125}+4\sqrt{20}=23\sqrt{5}$    

Hence, the last option i.e. $23\sqrt{5}$ is correct.                      

Step-by-step explanation:

Given the expression

$3\sqrt{125}+4\sqrt{20}$

Combining the radical expressions

$3\sqrt{125}+4\sqrt{20}$

let us first solve

$3\sqrt{125}$

$=3\sqrt{5^3}$

$=3\sqrt{5^2\cdot \:5}$

Apply radical rule:  $\sqrt{ab}=\sqrt{a}\sqrt{b},\:\quad \:a\ge 0,\:b\ge 0$

$=3\sqrt{5^2}\sqrt{5}$

Apply radical rule:   $\sqrt{a^2}=a,\:\quad \:a\ge 0$

$=3\cdot \:5\sqrt{5}$

$=15\sqrt{5}$

Thus,

$3\sqrt{125}=15\sqrt{5}$

similarly solving

$4\sqrt{20}$

$=4\sqrt{2^2\cdot \:5}$

Apply radical rule:  $\sqrt{ab}=\sqrt{a}\sqrt{b},\:\quad \:a\ge 0,\:b\ge 0$

$=4\sqrt{2^2}\sqrt{5}$

Apply radical rule:   $\sqrt{a^2}=a,\:\quad \:a\ge 0$

$=4\cdot \:2\sqrt{5}$

$=8\sqrt{5}$

Thus,

$4\sqrt{20}=8\sqrt{5}$

so we get

$3\sqrt{125}=15\sqrt{5}$

$4\sqrt{20}=8\sqrt{5}$

so the expression becomes

$3\sqrt{125}+4\sqrt{20}=15\sqrt{5}+8\sqrt{5}$        ∵ $3\sqrt{125}=15\sqrt{5}$ , $4\sqrt{20}=8\sqrt{5}$

                        $=23\sqrt{5}$                   ∵ $15\sqrt{5}+8\sqrt{5}=23\sqrt{5}$

Therefore, we conclude that:

$3\sqrt{125}+4\sqrt{20}=23\sqrt{5}$    

Hence, the last option i.e. $23\sqrt{5}$ is correct.