Solve Nested Roots: Simplifying ¹⁰√(⁵√100) Step by Step

When you have a root of a root, you're applying two operations in sequence. The mathematical property states $\sqrt[m]{\sqrt[n]{a}} = \sqrt[m \times n]{a}$. Think of it like exponents: $(a^{1/n})^{1/m} = a^{1/n \times 1/m} = a^{1/(mn)}$.

Yes! Since $100 = 10^2$, you can write $\sqrt[50]{100} = \sqrt[50]{10^2} = 10^{2/50} = 10^{1/25} = \sqrt[25]{10}$. This is the most simplified form.

The same rule applies! Multiply all the indices: $3 \times 4 \times 5 = 60$, so the answer is $\sqrt[60]{x}$. Work from inside out or multiply all at once.

Think of roots as fractional exponents: $\sqrt[10]{\sqrt[5]{100}} = (100^{1/5})^{1/10} = 100^{1/5 \times 1/10} = 100^{1/50} = \sqrt[50]{100}$. When you raise a power to another power, you multiply the exponents!

No! $\sqrt[10]{\sqrt[5]{100}}$ equals $\sqrt[5]{\sqrt[10]{100}}$ because multiplication is commutative: $10 \times 5 = 5 \times 10 = 50$.