1. Is the product of two irrational numbers always irrational? Justify your answer.
Answers
the given question we can say that the product of two irrational numbers are not always irrational. Note: ... The result may be rational or irrational but always it is not irrational.
Hope it helps:)
Answer:
Answer :-
The value of \sf x^{3} +\sf\cfrac{1}{{x}^{3}}x
3
+
x
3
1
= 488
To Find :-
The value of \sf x^{3} +\cfrac{1}{{x}^{3}}x
3
+
x
3
1
Given :-
\sf x + \cfrac{1}{x} = 8x+
x
1
=8
Step By Step Explanation :-
We know that \sf{x + \cfrac{1}{x} = 8}x+
x
1
=8
We need to calculate the value of \sf x^{3} +\cfrac{1}{{x}^{3}}x
3
+
x
3
1
So let's do it !!
\begin{gathered} \bf \: Using \: Identity \downarrow \\ \\\dag\boxed{ \bf{ \red {{(x + y)}^{3}= {x}^{3} + {y}^{3} + 3xy(x + y)}}}\end{gathered}
UsingIdentity↓
†
(x+y)
3
=x
3
+y
3
+3xy(x+y)
By substituting the values ⤵
\begin{gathered} \sf \left(\cfrac{1}{x}+x\right)^{3}= \cfrac{1}{ {x}^{3} } + {x}^{3} + 3 \times \cfrac{1}{ \not x} \times \not x \: \left( x + \cfrac{1}{x}\right) \\ \\ \bf \: By \: substituting \: the \: values \downarrow \\ \\\implies \sf {(8)}^{3} = \cfrac{1}{ {x}^{3} } + {x}^{3} + 3(8) \\ \\\implies \sf 512 = \cfrac{1}{ {x}^{3} } + {x}^{3} = \cfrac{1}{ {x}^{3} } + {x}^{3} \\ \\ \implies\bf 488 = \cfrac{1}{ {x}^{3} } + {x}^{3} \end{gathered}
(
x
1
+x)
3
=
x
3
1
+x
3
+3×
x
1
×
x(x+
x
1
)
Bysubstitutingthevalues↓
⟹(8)
3
=
x
3
1
+x
3
+3(8)
⟹512=
x
3
1
+x
3
+24
⟹512−24=
x
3
1
+x
3
⟹488=
x
3
1
+x
3
Hence the value of \sf x^{3} +\cfrac{1}{{x}^{3}}x
3
+
x
3
1
= 488
__________________________