solve the equation m^6 -64=0
M^6-64=0
M^6=64
take 6root of M^6 to get M
take 6root of 64 which =+or-2
so,M=+-2
(m^2-4)(m^4+4m^2+16) = 0
(m+2)(m-2)(m^4+4m^2+16) = 0
so, m = 2 and -2 are solutions
Now we have a quadratic in m^2, so
m^2 = [-4 +/- sqrt(-48)]/2
= -2 +/- 4sqrt(3)i
m = +/- 1 +/- sqrt(3)i
To solve the equation m^6 - 64 = 0, we need to find the values of m that make this equation true.
Step 1: Begin by factoring the equation. We can rewrite 64 as 2^6, so the equation becomes m^6 - 2^6 = 0, which can be factored as (m^3)^2 - (2^3)^2 = 0.
Step 2: Apply the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b). In our equation, let a = m^3 and b = 2^3. Therefore, we have (m^3 + 2^3)(m^3 - 2^3) = 0.
Step 3: Simplify each factor separately.
- For the first factor, m^3 + 2^3, we observe that (m + 2)(m^2 - 2m + 4) = 0 by applying the sum of cubes formula. However, this factor does not lead to any real solutions since there are no real numbers that make m^2 - 2m + 4 = 0.
- For the second factor, m^3 - 2^3, we can apply the difference of cubes formula: (m - 2)(m^2 + 2m + 4) = 0.
Step 4: Solve each factor separately.
- From the first factor, m + 2 = 0, we find that m = -2.
- From the second factor, m - 2 = 0, we find that m = 2.
Therefore, the solutions to the equation m^6 - 64 = 0 are m = -2 and m = 2.