d,$\sqrt[]{12+2.\sqrt[]{27}}=\sqrt[]{9+2.\sqrt[]{27}+3}=\sqrt[]{(\sqrt[]9+\sqrt[]3)^2}=\sqrt[]9+\sqrt[]3$

e,$\sqrt[]{23-2.\sqrt[]120}=\sqrt[]{15+\sqrt[]2.\sqrt[]{15}.\sqrt[]8+8}=\sqrt[]{(\sqrt[]{15}+\sqrt[]8)^2}=\sqrt[]{15}=\sqrt[]8$

 f, $\sqrt[]{2.\sqrt[]{84}+20}=\sqrt[]{14+2\sqrt[]{14}.\sqrt[]6+6}=\sqrt[]{(\sqrt[]{14}+\sqrt[]6)^2}=\sqrt[]{14}+\sqrt[]6$

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)

$\sqrt[]{16-2\sqrt[]{55}}$

= $\sqrt[]{11-2\sqrt[]{11}.\sqrt[]{5}+5}$

= $\sqrt[]{(\sqrt[]{11}-\sqrt[]{5})^2}$

= | $\sqrt[]{11}$ - $\sqrt[]{5}$ |

= $\sqrt[]{11}$ - $\sqrt[]{5}$