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If someone asked you to simplify 49\sqrt{49}, you'd instantly say 77. But what if you saw x9\sqrt{x^9}?
Every radical has an index (the small number indicating the root type) and a radicand (the entire expression trapped under the root symbol).
How do we pull variable powers out of that root symbol cleanly every single time?

The Exponent Division Rule

Since squaring multiplies an exponent by 22 through the power-of-a-power rule (x3)2=x6(x^3)^2 = x^6, taking a square root does the exact opposite by dividing the exponent by 22.
What happens when the variable's exponent inside the radical is an odd number that doesn't divide evenly by 2?

Pulling Out Perfect Squares

When simplifying a square root, we separate each variable into the largest even exponent and the leftover single factor.
The foundation for breaking apart roots is the Product Property of Radicals, which states that a⋅bn=an⋅bn\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} for real numbers where the roots exist.
This rule allows us to split any radicand into factors that are perfect powers and factors that remain trapped inside the radical.
The Product Property of Radicals states that a⋅bn=an⋅bn\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} for real numbers where the roots exist.