if x2 + 1/x2 =8 3/5 (in mixed fraction) then find the value of x3 +1/125x3
Answers
Answer:
Answer:
\begin{gathered}{x}^{3} + \frac{1}{125 {x}^{3} } = ± \frac{126}{5} \\\end{gathered}
x
3
+
125x
3
1
=±
5
126
Step-by-step explanation:
If
\begin{gathered}{x}^{2} + \frac{1}{25 {x}^{2} } = \frac{43}{5} \\ \\\end{gathered}
x
2
+
25x
2
1
=
5
43
To find the value of
{x}^{3} + \frac{1}{125 {x}^{3} }x
3
+
125x
3
1
First find the value of
\begin{gathered}(x + \frac{1}{5x} ) \\ \\ for \: that \\ convert \: {x}^{2} + \frac{1}{25 {x}^{2} } \: into \: complete \: square \\ \\ {x}^{2} + 2x \frac{1}{5x} + \frac{1}{25 {x}^{2} } = \frac{43}{5} + \frac{2}{5} \\ \\ {(x + \frac{1}{5x}) }^{2} = \frac{45}{5} \\ \\ {(x + \frac{1}{5x}) }^{2} = {(3)}^{2} \\ \\ x + \frac{1}{5x} = ± 3 \\ \\\end{gathered}
(x+
5x
1
)
forthat
convertx
2
+
25x
2
1
intocompletesquare
x
2
+2x
5x
1
+
25x
2
1
=
5
43
+
5
2
(x+
5x
1
)
2
=
5
45
(x+
5x
1
)
2
=(3)
2
x+
5x
1
=±3
So,
\begin{gathered}{x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} ) \\ \\ {x}^{3} + ({ { \frac{1}{5x} })^{3} } = (x + \frac{1}{5x} )( {x}^{2} - x( \frac{1}{5x} ) + \frac{1}{25 {x}^{2} } \\ \\ = 3( \frac{43}{5} - \frac{1}{5} ) \\ \\ = 3 \times \frac{43 - 1}{5} \\ \\ {x}^{3} + \frac{1}{125 {x}^{3} }= \frac{126}{5} \\ \\ or \\ \\ {x}^{3} + \frac{1}{125 {x}^{3} } = - \frac{126}{5}\end{gathered}
x
3
+y
3
=(x+y)(x
2
−xy+y
2
)
x
3
+(
5x
1
)
3
=(x+
5x
1
)(x
2
−x(
5x
1
)+
25x
2
1
=3(
5
43
−
5
1
)
=3×
5
43−1
x
3
+
125x
3
1
=
5
126
or
x
3
+
125x
3
1
=−
5
126