sqrt(sqrt[4](x)+7)+sqrt(x)-7=0

To solve the equation \( \sqrt{\sqrt[4]{x}+7}+\sqrt{x}-7=0 \), we can start by isolating one of the square roots. Let's isolate \( \sqrt{\sqrt[+7} \):<br /><br />\( \sqrt{\sqrt[4]{x}+7} = -\sqrt{x}+7 \)<br /><br />Now, square both sides to eliminate the square root:<br /><br />\( \sqrt[4]{x}+7 = (\sqrt{x}-7)^2 \)<br /><br />\( \sqrt[4]{x}+7 = x-14\sqrt \)<br /><br />Next, let's isolate \( \sqrt[4]{x} \):<br /><br />\( \sqrt[4]{x} = x-14\sqrt{x}+42 \)<br /><br />Now, raise both sides to the power of 4 to eliminate the fourth root:<br /><br />\( x = (x-14\sqrt{x}+42)^4 \)<br /><br />This equation is quite complex to solve directly. However, we can try to find a solution by substituting some values for \( x \) and checking if they satisfy the equation.<br /><br />Let's try \( x = 49 \):<br /><br />\( \sqrt{\sqrt[4]{49}+7}+\sqrt{49}-7 = \sqrt{8+7}+7-7 = \sqrt{15}+7-7 = \sqrt{15} \neq 0 \)<br /><br />So, \( x = 49 \) is not a solution.<br /><br />Let's try \( x = 36 \):<br /><br />\( \sqrt{\sqrt[4]{36}+7}+\sqrt{36}-7 = \sqrt{3+7}+6-7 = \sqrt{10}+6-7 = \sqrt{10}-1 \neq 0 \)<br /><br />So, \( x = 36 \) is not a solution.<br /><br />Let's try \( x = 16 \):<br /><br />\( \sqrt{\sqrt[4]{16}+7}+\sqrt{16}-7 = \sqrt{2+7}+4-7 = \sqrt{9}+4-7 = 3+4-7 = 0 \)<br /><br />So, \( x = 16 \) is a solution.<br /><br />Therefore, the solution to the equation \( \sqrt{\sqrt[4]{x}+7}+\sqrt{x}-7=0 \) is \( x = 16 \).