1Cademy - Simplifying Variable Roots with Even Indices

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Properties of $\sqrt[n]{a}$

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Simplifying Variable Roots with Even Indices

When simplifying a radical where the index $n$ is even, the property $\sqrt[n]{a^n} = |a|$ must be applied to ensure the principal root is non-negative. This concept extends to expressions with variables raised to higher powers, following the pattern $\sqrt{a^{2m}} = |a^m|$ . If the resulting exponent of the variable is odd, an absolute value sign is required around the variable. However, if the resulting expression is inherently non-negative (e.g., raised to an even power), absolute value signs can be omitted; for instance, $\sqrt[4]{z^8} = \sqrt[4]{(z^2)^4} = z^2$ since $z^2$ is always positive. When a numerical coefficient is present, its principal root is computed normally, and absolute value is applied only to the variable portion (e.g., $\sqrt{16n^2} = \sqrt{(4n)^2} = 4|n|$ ). Additionally, if a negative sign is outside the radical, it is applied after evaluating the root, such as $-\sqrt{81c^2} = -9|c|$ .

Updated 2026-05-01

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