If-x-is-real-number-satisfying-3x-1-2x-4-find-the-value-of-27x-3-1-8x-3-

Question Number 119692 by bemath last updated on 26/Oct/20

I f x i s r e a l n u m b e r s a t i s f y i n g

3 x + \frac{1}{2 x} = 4, f i n d t h e v a l u e o f

27 x^{3} + \frac{1}{8 x^{3}} .

Commented by bemath last updated on 26/Oct/20

y e s g a v e k u d o s

Answered by $@y@m last updated on 26/Oct/20

U s e f o r m u l a

a^{3} + b^{3} = {(a + b)}^{3} - 3 a b (a + b)

a n d t r y y o u r s e l f .

A n s : 46

Answered by Ar Brandon last updated on 26/Oct/20

3 x + \frac{1}{2 x} = 4 \Rightarrow {(3 x + \frac{1}{2 x})}^{2} = 16

\Rightarrow {9 x}^{2} + 3 + \frac{1}{{4 x}^{2}} = 16 \Rightarrow {9 x}^{2} + \frac{1}{{4 x}^{2}} = 13

\Rightarrow ({9 x}^{2} + \frac{1}{{4 x}^{2}}) (3 x + \frac{1}{2 x}) = 13 \times 4

\Rightarrow {27 x}^{3} + \frac{9 x}{2} + \frac{3}{4 x} + \frac{1}{{8 x}^{3}} = 52

\Rightarrow ({27 x}^{3} + \frac{1}{{8 x}^{3}}) = 52 - \frac{3}{2} (3 x + \frac{1}{2 x}) = 52 - \frac{3 \times 4}{2}

\Rightarrow {27 x}^{3} + \frac{1}{{8 x}^{3}} = 46

Answered by mathmax by abdo last updated on 26/Oct/20

(3 x + \frac{1}{2 x}) = 4 \Rightarrow {(3 x + \frac{1}{2 x})}^{3} = 64 \Rightarrow

{27 x}^{3} + 3 . (3 x) . \frac{1}{2 x} (3 x + \frac{1}{2 x}) + \frac{1}{{8 x}^{3}} = 64 \Rightarrow

{27 x}^{3} + \frac{1}{{8 x}^{3}} + 18 = 64 \Rightarrow {27 x}^{3} + \frac{1}{{8 x}^{3}} = 64 - 18 = 46