A48
B52
C42
D54
Answer:
B. 52
Read Explanation:
Algebraic Simplification and Identities
Understanding the Problem
The question involves finding the value of an expression involving powers of a variable and its reciprocal, given a relationship between the variable and its reciprocal.
Key Algebraic Identity
The core identity used here is the sum of cubes formula: (a + b)³ = a³ + b³ + 3ab(a + b)
Applying the Identity
In this problem, let a = m and b = 1/m.
Substituting these into the identity: (m + 1/m)³ = m³ + (1/m)³ + 3(m)(1/m)(m + 1/m)
This simplifies to: (m + 1/m)³ = m³ + 1/m³ + 3(m + 1/m)
Using the Given Information
We are given that m + 1/m = 4.
Substitute the given value into the simplified identity: (4)³ = m³ + 1/m³ + 3(4)
Calculation
Calculate the cube of 4: 4³ = 64
Calculate the product of 3 and 4: 3 * 4 = 12
The equation becomes: 64 = m³ + 1/m³ + 12
Isolating the Target Expression
To find the value of m³ + 1/m³, rearrange the equation:
m³ + 1/m³ = 64 - 12
m³ + 1/m³ = 52