2. If x + 1/x = 1, then the value of x6 + 1/x6 is:
(A) −2
(B) −1
(C) 1
(D) 2
If x + 1/x = 1, Find x⁶ + 1/x⁶: Detailed Algebraic Solution
Correct Option: (D) 2
Understanding the Given Algebraic Expression
We are given the important algebraic relation x + 1/x = 1 and asked to calculate the value of x6 + 1/x6. At first sight, directly finding the sixth power may appear lengthy because calculating x explicitly and then raising it to the sixth power seems complicated. However, the expression has a highly symmetric structure involving a number and its reciprocal, so algebraic identities provide a much simpler solution.
The main idea is to begin with the given expression and gradually calculate higher powers. Since the required expression contains the sixth power, we can first determine the second power, then the third power, and finally use the third-power result to reach the sixth power.
Step-by-Step Solution Using Algebraic Identities
Finding x² + 1/x²
The given condition is:
x + 1/x = 1
Squaring both sides gives:
(x + 1/x)2 = 12
Using the identity (a + b)2 = a2 + 2ab + b2, we obtain:
x2 + 2(x)(1/x) + 1/x2 = 1
Since x × 1/x = 1, the middle term becomes 2. Therefore,
x2 + 2 + 1/x2 = 1
Subtracting 2 from both sides gives:
x2 + 1/x2 = −1
Thus, the sum of the squares of x and its reciprocal is −1. This intermediate result will help us calculate the third powers.
Finding x³ + 1/x³
Now consider the product:
(x + 1/x)(x2 + 1/x2)
Multiplying the expressions gives:
x3 + x/x2 + x2/x + 1/x3
After simplifying the middle terms, we get:
x3 + x + 1/x + 1/x3
Therefore,
(x + 1/x)(x2 + 1/x2)
=
x3 + 1/x3 + x + 1/x
From the given condition, x + 1/x = 1, and from the previous calculation, x2 + 1/x2 = −1. Substituting these values gives:
(1)(−1) = x3 + 1/x3 + 1
−1 = x3 + 1/x3 + 1
Subtracting 1 from both sides gives:
x3 + 1/x3 = −2
Finding x⁶ + 1/x⁶
We have now reached the most useful intermediate result:
x3 + 1/x3 = −2
Squaring both sides gives:
(x3 + 1/x3)2 = (−2)2
Expanding the left-hand side,
x6 + 2(x3)(1/x3) + 1/x6 = 4
Since x3 × 1/x3 = 1, we obtain:
x6 + 2 + 1/x6 = 4
Subtracting 2 from both sides gives:
x6 + 1/x6 = 2
Therefore, the required value is 2, and the correct answer is Option (D).
Alternative Direct Method Using the Original Equation
The same question can also be solved through an elegant direct approach. Starting with:
x + 1/x = 1
Since x cannot be zero, multiply the entire equation by x:
x2 + 1 = x
Rearranging gives:
x2 − x + 1 = 0
From x2 − x + 1 = 0, we have:
x2 = x − 1
Multiplying both sides by x gives:
x3 = x2 − x
Substituting x2 = x − 1 gives:
x3 = (x − 1) − x
x3 = −1
Squaring both sides gives:
x6 = 1
Taking the reciprocal also gives:
1/x6 = 1
Therefore,
x6 + 1/x6 = 1 + 1 = 2
This direct method confirms the same answer and provides a quick way to solve the problem once the relation x3 = −1 is recognized.
Detailed Explanation of Every Option
Option (A): −2
Option (A) is incorrect. The value −2 is actually obtained for the intermediate expression x3 + 1/x3, not for the required sixth-power expression. Since x3 = −1, squaring gives x6 = 1. Therefore, the sixth-power expression becomes positive.
Option (B): −1
Option (B) is incorrect. The value −1 corresponds to the expression x2 + 1/x2. It is obtained by squaring the original condition and subtracting the middle term 2. However, the question asks for the sixth powers, so −1 is only an intermediate result and not the final answer.
Option (C): 1
Option (C) is incorrect. Although x6 = 1, the required expression contains two terms: x6 and 1/x6. Both terms are equal to 1, so they must be added. Therefore, the final result is 1 + 1 = 2, not 1.
Option (D): 2
Option (D) is correct. The given condition x + 1/x = 1 leads to x2 − x + 1 = 0, from which x3 = −1. Squaring this relation gives x6 = 1, and consequently 1/x6 = 1. Therefore, x6 + 1/x6 = 2.
Final Answer
Starting from x + 1/x = 1, we find that x2 − x + 1 = 0. This relation leads to x3 = −1 and therefore x6 = 1. Since the reciprocal sixth power is also equal to 1, the required expression is:
x6 + 1/x6 = 1 + 1 = 2
Correct Option: (D) 2
