SOLUTION: Let P = 2^{1/2} \cdot 4^{1/4} \cdot 8^{1/8} \cdot 16^{1/16} Then P can be expressed in the form \sqrt[a]{b}, where $a$ and $b$ are positive integers. Find the smallest possible

Question 1209816: Let
P = 2^{1/2} \cdot 4^{1/4} \cdot 8^{1/8} \cdot 16^{1/16}
Then P can be expressed in the form \sqrt[a]{b}, where $a$ and $b$ are positive integers. Find the smallest possible value of $a + b.$
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let's rewrite the expression for $P$ using powers of 2:
$$P = 2^{1/2} \cdot (2^2)^{1/4} \cdot (2^3)^{1/8} \cdot (2^4)^{1/16}$$
$$P = 2^{1/2} \cdot 2^{2/4} \cdot 2^{3/8} \cdot 2^{4/16}$$
$$P = 2^{1/2} \cdot 2^{1/2} \cdot 2^{3/8} \cdot 2^{1/4}$$
Now, let's add the exponents:
$$P = 2^{1/2 + 1/2 + 3/8 + 1/4} = 2^{1 + 3/8 + 2/8} = 2^{1 + 5/8} = 2^{13/8}$$
We want to express $P$ in the form $\sqrt[a]{b}$, which means $P = b^{1/a}$.
So we have $P = 2^{13/8} = (2^{13})^{1/8}$.
Thus, we can write $P = \sqrt[8]{2^{13}}$.
In this case, $a = 8$ and $b = 2^{13} = 8192$.
Then $a + b = 8 + 8192 = 8200$.
We want to find the smallest possible value of $a + b$.
Let $P = 2^{13/8}$. We can write this as $P = (2^{13k})^{1/(8k)}$ for any positive integer $k$.
When $k=1$, $a=8$ and $b=2^{13} = 8192$, so $a+b = 8200$.
When $k=2$, $a=16$ and $b=2^{26}$, so $a+b = 16 + 2^{26} = 16 + 67108864 = 67108880$.
Since $2^{13/8}$ is already in its simplest form, we cannot reduce the fraction 13/8 further. So the smallest possible value for $a$ is 8.
Thus, the smallest possible value of $a + b$ is $8 + 2^{13} = 8 + 8192 = 8200$.
Final Answer: The final answer is $\boxed{8200}$

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