Help me with this math problem?

Question

Find log ((20+14*sqrt(2))^(1/3)+(20-14*sqrt(2))^(1/3)) base 2.

Please show all your work step by step and tell me the right answer. The base 2 is lower case, subscript. Thanks.

Answer 1

(20 + 14 * sqrt(2))^(1/3) + (20 - 14 * sqrt(2))^(1/3)



Cube everything



a = (20 + 14 * sqrt(2))^(1/3)

b = (20 - 14 * sqrt(2))^(1/3)



(a + b)^3 =>

a^3 + 3a^2 * b + 3ab^2 + b^3 =>

a^3 + b^3 + 3ab * (a + b) =>

20 + 14 * sqrt(2) + 20 - 14 * sqrt(2) + 3 * ((20 + 14 * sqrt(2)) * (20 - 14 * sqrt(2)))^(1/3) * ((20 + 14 * sqrt(2))^(1/3) + (20 + 14 * sqrt(2))^(1/3)) =>

40 + 3 * (400 - 196 * 2)^(1/3) * (a + b) =>

40 + 3 * (400 - 392)^(1/3) * (a + b) =>

40 + 3 * 8^(1/3) * (a + b) =>

40 + 3 * 2 * (a + b) =>

40 + 6 * (a + b)



(a + b)^3 = 6 * (a + b) + 40



a + b = x



x^3 = 6x + 40

x^3 - 6x - 40 = 0



Try the rational root theorem to find solutions for x



x = -40 , -20 , -10 , -8 , -5 , -4 , -2 , -1 , 1 , 2 , 4 , 5 , 8 , 10 , 20 , 40

x = 1 : 1 - 6 - 40 = 1 - 46 = -45

x = 2 : 8 - 12 - 40 = 8 - 52 = -44

x = 4 : 64 - 24 - 40 = 64 - 64 = 0



x = 4 is a solution, so x - 4 is a factor



(x - 4) * (ax^2 + bx + c) = x^3 - 6x - 40

ax^3 + bx^2 + cx - 4ax^2 - 4bx - 4c = x^3 + 0x^2 - 6x - 40



a = 1

b - 4a = 0

b = 4a

b = 4



c - 4b = -6

c - 16 = -6

c = 10



-4c = -40

-40 = -40



(x - 4) * (x^2 + 4x + 10)

(x - 4) * (x^2 + 4x + 4 + 6)

(x - 4) * ((x + 2)^2 + 6)



(x + 2)^2 + 6 = 0

(x + 2)^2 = -6

No real solution



So



(20 + 14 * sqrt(2))^(1/3) + (20 - 14 * sqrt(2))^(1/3) = 4



Now we have:



log[2](4) =>

log[2](2^2) =>

2 * log[2](2) =>

2 * 1 =>

2

Answer 2

log2(∛(20 +14√2) +∛(20 -14√2))

≈ log2(∛39.7989898732 +∛0.20101012678)

≈ log2(3.41421356237 +0.58578643763)

= log2(4)

= 2

Answer 3

The expression is not displaying completely because it is too long. Insert a few blank spaces to break it up into smaller strings, then post it as a new question.



log₂((20+14√2)^⅓ + (20-14√2)^⅓) = log₂(4) = 2

Answer 4

I get 2.

It's just arithmetic, do it step by step.

Answer 5

log ((20+14*sqrt(2))^(1/3)+(20-14*sqrt(2))^(1/3)) base 2



what is base 2?



is this it?

log₂ ((∛(20+14•√2)) + ∛(20–14•√2))

log₂ (3.414214+ 0.2010101)

log₂ 3.213204

1.684013