Question

√(√32+x)√(√32+x)=x

Answer 1

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.