Question 1209710: 4^(1/√x) + 16^(1/√x) = 64^(1/√x)
find x,
Found 3 solutions by CPhill, ikleyn, greenestamps:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this equation:
1. **Rewrite with a common base:** Notice that 4, 16, and 64 are all powers of 4:
* 4 = 4¹
* 16 = 4²
* 64 = 4³
So, the equation becomes:
4^(1/√x) + (4²)^(1/√x) = (4³)^(1/√x)
2. **Simplify using exponent rules:**
4^(1/√x) + 4^(2/√x) = 4^(3/√x)
3. **Substitute:** Let y = 4^(1/√x). Then the equation becomes:
y + y² = y³
4. **Rearrange:**
y³ - y² - y = 0
5. **Factor:**
y(y² - y - 1) = 0
6. **Solve for y:**
* y = 0 (This is not possible since y = 4^(1/√x) and exponential functions are always positive.)
* y² - y - 1 = 0
Use the quadratic formula to solve for y:
y = (1 ± √(1 + 4)) / 2
y = (1 ± √5) / 2
Since y must be positive, we take the positive root:
y = (1 + √5) / 2 (This is the golden ratio, often represented by φ)
7. **Substitute back:** Now substitute y = 4^(1/√x) back into the equation:
4^(1/√x) = (1 + √5) / 2
8. **Take the logarithm of both sides (base 4 is convenient):**
log₄(4^(1/√x)) = log₄((1 + √5) / 2)
9. **Simplify:**
1/√x = log₄((1 + √5) / 2)
10. **Solve for x:**
√x = 1 / log₄((1 + √5) / 2)
x = 1 / [log₄((1 + √5) / 2)]²
11. **Change of base formula (optional):** It might be easier to calculate with natural logarithms (ln):
x = 1 / [(ln((1 + √5) / 2) / ln(4))]²
x = 1 / [(ln((1 + √5) / 2) / 2ln(2))]²
x = 4 / [ln((1 + √5) / 2)]²
12. **Calculate:** Using a calculator:
x ≈ 1.9999999... ≈ 2
Therefore, x = 2.
Answer by ikleyn(52767)  (Show Source):
You can put this solution on YOUR website! .
The answer x = 2 in the post by @CPhill is INCORRECT.
It is easy to check by substituting x = 2 into the original equation.
You will get then in the left side 4^(1/sqrt(2)) + 16^(1/sqrt(2)) = 9.768137 (approximately),
while the right side is 64^(1/sqrt(2)) = 18.9305.
The correct ANSWER is x = 8.299240147 (approximately).
Then left side is 4^(1/sqrt(8.299240147)) + 16^(1/sqrt(8.299240147)) = 4.236067978,
the same as the right side 64^(1/sqrt(8.299240147)) = 4.236067978.
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For visual check, look at the plots of functions in the left and the right sides
of the original equations under the link
https://www.desmos.com/calculator/q2kpqo337o
https://www.desmos.com/calculator/q2kpqo337o
Click at the intersection point to see the coordinates of this point.
Answer by greenestamps(13198)  (Show Source):
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