Solve the Nested Radical: Fourth Root of Square Root of 6

Think of it as applying operations in sequence. First you take the square root (power of 1/2), then the fourth root (power of 1/4). When you raise a power to another power, you multiply the exponents: $(6^{1/2})^{1/4} = 6^{1/2 \times 1/4} = 6^{1/8}$

Those are completely different expressions! $\sqrt[4]{\sqrt{6}}$ means you're taking the fourth root of the result of the square root. Multiplication would give you $6^{1/2} \times 6^{1/4} = 6^{3/4}$, which is much larger.

Since 6 = 2 × 3 and neither 2 nor 3 are perfect eighth powers, $\sqrt[8]{6}$ is already in its simplest radical form. You could write it as a decimal approximation, but the radical form is exact.

Apply the same rule! Multiply all three indices: $3 \times 4 \times 2 = 24$, so the answer would be $\sqrt[24]{5}$. Work from the innermost radical outward.

Yes, absolutely! $\sqrt[4]{\sqrt{6}}$ is different from $\sqrt{\sqrt[4]{6}}$. Always work from the inside out, applying the innermost operation first.