1Cademy - Simplifying $$\sqrt{\frac{a^8}{a^6}}$$ and $$\sqrt{\frac{x^{14}}{x^{10}}}$$

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Simplifying $\sqrt{\frac{m^6}{m^4}}$

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Simplifying $\sqrt{\frac{a^8}{a^6}}$ and $\sqrt{\frac{x^{14}}{x^{10}}}$

Simplify two square roots whose radicands are fractions of variables with like bases by first applying the Quotient Property for Exponents.

ⓐ $\sqrt{\frac{a^8}{a^6}}$ : Step 1 — Simplify the fraction inside the radical. Both terms share the base $a$ . Subtract the exponents: $\frac{a^8}{a^6} = a^{8-6} = a^2$ . Step 2 — Simplify the square root. Evaluate the square root: $\sqrt{a^2} = |a|$ .

ⓑ $\sqrt{\frac{x^{14}}{x^{10}}}$ : Step 1 — Simplify the fraction inside the radical. Both terms share the base $x$ . Subtract the exponents: $\frac{x^{14}}{x^{10}} = x^{14-10} = x^4$ . Step 2 — Simplify the square root. Evaluate the square root. Since $x^4 = (x^2)^2$ , taking the principal square root gives $x^2$ .

In both cases, reducing the fraction under the radical first produces a perfect square that can be easily simplified. Note that an absolute value is required for $|a|$ because the root's index is even and the resulting exponent is odd.

Updated 2026-05-26

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OpenStax Intermediate Algebra

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