if-xyz-2-x-y-z-5-then-find-x-3-y-3-z-3-x-y-z-x-2-y-2-z-2-

Question Number 147187 by mathdanisur last updated on 18/Jul/21

i f {(\overset{―}{x y z})}^{2} = {(x + y + z)}^{5} t h e n f i n d :

x^{3} + y^{3} + z^{3} - ∣ (x + y + z) + (x^{2} + y^{2} + z^{2}) ∣

Commented by Rasheed.Sindhi last updated on 18/Jul/21

I s \overset{―}{x y z} a d i g i t a l r e p r e s e n t a t i o n o f

d e c i m a l n u m b e r ?

Commented by mathdanisur last updated on 18/Jul/21

Y e s S e r

Answered by Rasheed.Sindhi last updated on 19/Jul/21

{(\overset{―}{x y z})}^{2} = {(x + y + z)}^{5}

▸ \overset{―}{x y z} i s 3 - d i g i t n a t u r a l n u m b e r

▸ {(\overset{―}{x y z})}^{2} i s p e r f e c t 5 t h p o w e r

\Rightarrow \overset{―}{x y z} i s p e r f e c t 5 t h p o w e r

T h e o n l y t h r e e d i g i t n u m b e r w h i c h

i s a l s o 5 t h p o w e r i s 243 (= 3^{5})

[2^{5} = 32 2 - d i g i t n u m b e r \times

3^{5} = 243 3 - d i g i t n u m b e r ✓

4^{5} = 1024 4 - d i g i t n u m b e r \times]

P o s s i b l e c a n d i d a t e i s o n l y 243 a n d

i t a l s o f u l f i l l s t h e g i v e n c o n d i t i o n :

{(243)}^{2} \overset{?}{=} {(2 + 4 + 3)}^{5}

{(3^{5})}^{2} \overset{?}{=} 9^{5}

{(3^{5})}^{2} \overset{?}{=} {(3^{2})}^{5}

3^{10} = 3^{10}

∴ 243 i s o n l y s u c c e s s f u l c a n d i d a t e .

∴ x = 2, y = 4, z = 3

x^{3} + y^{3} + z^{3} - ∣ (x + y + z) + (x^{2} + y^{2} + z^{2}) ∣

= 2^{3} + 4^{3} + 3^{3} - ∣ (2 + 4 + 3) + (2^{2} + 3^{2} + 4^{2} ∣

= 8 + 64 + 27 - ∣ 9 + 29 ∣

= 61

Commented by mathdanisur last updated on 19/Jul/21

c o o l S e r t h a n k s