To simplify the expression, we can start by simplifying the inner square roots.
√(32 + x) can be written as √(16 * 2 + x), which further simplifies to √(16) * √(2 + x), giving us 4√(2 + x).
So our equation becomes:
4√(2 + x) * √(4√(2 + x)) = x
Now, we can simplify the square root of the square root by raising the expression to the power of 1/4:
(4√(2 + x))^(1/4) * (√(4√(2 + x)))^(1/2) = x
Simplifying further:
(4√(2 + x))^(1/4) * (2√(2 + x))^(1/2) = x
Now, let's eliminate the square roots by raising both sides of the equation to the power of 4:
[(4√(2 + x))^(1/4) * (2√(2 + x))^(1/2)]^4 = x^4
Simplifying further:
[4^(1/4) * √(2 + x)^(1/4) * 2^(1/2) * (2 + x)^(1/2)]^4 = x^4
Now, let's simplify the constants outside the square roots:
(4^(1/4) * 2^(1/2))^4 * [√(2 + x)^(1/4) * (2 + x)^(1/2)]^4 = x^4
16 * [√(2 + x)^(1/4 + 1) * (2 + x)^(1/2 + 1)]^4 = x^4
16 * [√(2 + x)^(5/4) * (2 + x)^(3/2)]^4 = x^4
Now, let's simplify the square root and powers of (2 + x):
16 * [(2 + x)^(5/4) * (2 + x)^(3/2)]^4 = x^4
16 * (2 + x)^[(5/4) * 4] * (2 + x)^[(3/2) * 4] = x^4
16 * (2 + x)^5 * (2 + x)^6 = x^4
Now, let's simplify the equation:
16 * (2 + x)^11 = x^4
To find the value of x that satisfies this equation, we can solve through numerical methods or approximation techniques.
√(√32+x)√(√32+x)=x
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