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If x + 1/x = 7, find the value of x³ + 1/x³.
322
343
301
364
322
For x+x1=k, the value of x3+x31 is simply k3−3k. Here, calculate 73−3(7)=343−21=322.
It is given that the sum of a number and its reciprocal is 7, i.e., x + 1/x = 7.
x3+x31=(x+x1)3−3(x+x1)
For x+x1=k, the value of x3+x31 is simply k3−3k. Here, calculate 73−3(7)=343−21=322.
Students often misremember the formula as k3+3k instead of k3−3k, leading to an incorrect result of 364.
Identify the given value
We are given x+x1=7. We need to evaluate the expression x3+x31.
x+x1=7
Use the cubic identity
Using the algebraic expansion of (a+b)3=a3+b3+3ab(a+b), we derive the formula x3+x31=(x+x1)3−3(x+x1).
x3+x31=(x+x1)3−3(x+x1)
Substitute and solve
Substitute k=7 into the formula: 73−3(7)=343−21=322.
343−21=322
A is correct because substituting k=7 into k3−3k yields 343−21=322.
This identity is a specific case of xn+1/xn problems often found in competitive exams. Remember k2−2 for squares and k3−3k for cubes to save time.
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