d)
$\sqrt[]{12+2\sqrt[]{27}}$
= $\sqrt[]{3+2\sqrt[]{3}.\sqrt[]{9}+9}$
= $\sqrt[]{(\sqrt[]{3}+3)^2}$
= | $\sqrt[]{3}$ + 3 |
= 3 + $\sqrt[]{3}$
e)
$\sqrt[]{23-2\sqrt[]{120}}$
= $\sqrt[]{15-2\sqrt[]{15}.\sqrt[]{8}+8}$
= $\sqrt[]{(\sqrt[]{15}-\sqrt[]{8})^2}$
= | $\sqrt[]{15}$ - $\sqrt[]{8}$ |
= $\sqrt[]{15}$ - 2$\sqrt[]{2}$
f)
$\sqrt[]{2\sqrt[]{84}+20}$
= $\sqrt[]{6+2\sqrt[]{6}.\sqrt[]{14}+14}$
= $\sqrt[]{(\sqrt[]{6}+\sqrt[]{14})^2}$
= | $\sqrt[]{6}$ + $\sqrt[]{14}$ |
= $\sqrt[]{6}$ + $\sqrt[]{14}$
n)
$\sqrt[]{10-2\sqrt[]{21}}$
= $\sqrt[]{3-2\sqrt[]{3}.\sqrt[]{7}+7}$
= $\sqrt[]{(\sqrt[]{3}-\sqrt[]{7})^2}$
= | $\sqrt[]{3}$ - $\sqrt[]{7}$ |
= $\sqrt[]{7}$ - $\sqrt[]{3}$
r)
$\sqrt[]{14-2\sqrt[]{33}}$
= $\sqrt[]{3-2\sqrt[]{3}.\sqrt[]{11}+11}$
= $\sqrt[]{(\sqrt[]{3}-\sqrt[]{11})^2}$
= | $\sqrt[]{3}$ - $\sqrt[]{11}$ |
= $\sqrt[]{11}$ - $\sqrt[]{3}$
t)
$\sqrt[]{16-2\sqrt[]{55}}$
= $\sqrt[]{11-2\sqrt[]{11}.\sqrt[]{5}+5}$
= $\sqrt[]{(\sqrt[]{11}-\sqrt[]{5})^2}$
= | $\sqrt[]{11}$ - $\sqrt[]{5}$ |
= $\sqrt[]{11}$ - $\sqrt[]{5}$
