1Cademy - Simplifying $$\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}$$ and $$\frac{\sqrt[4]{96x^7}}{\sqrt[4]{3x^2}}$$

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Simplifying $\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}$ and $\frac{\sqrt[4]{96x^7}}{\sqrt[4]{3x^2}}$

Simplify two quotients of higher roots by applying the Quotient Property of $n$ th Roots in reverse to combine each quotient into a single radical, then simplifying.

ⓐ $\frac{\sqrt[3]{-108}}{\sqrt[3]{2}}$ : Neither radicand is a perfect cube, so use the Quotient Property to combine both radicals under one radical sign: $\sqrt[3]{\frac{-108}{2}}$ . Simplify the fraction: $\frac{-108}{2} = -54$ . Rewrite $-54$ as a product using perfect cube factors: $-54 = (-3)^3 \cdot 2$ . Split into two radicals: $\sqrt[3]{(-3)^3} \cdot \sqrt[3]{2}$ . Since $\sqrt[3]{(-3)^3} = -3$ , the result is $-3\sqrt[3]{2}$ .

ⓑ $\frac{\sqrt[4]{96x^7}}{\sqrt[4]{3x^2}}$ : Neither radicand is a perfect fourth power, so use the Quotient Property to write as one radical: $\sqrt[4]{\frac{96x^7}{3x^2}}$ . Simplify the fraction under the radical: $\frac{96}{3} = 32$ and $\frac{x^7}{x^2} = x^5$ , giving $\sqrt[4]{32x^5}$ . Rewrite using perfect fourth power factors: $32x^5 = 2^4 x^4 \cdot 2x$ . Split into two radicals: $\sqrt[4]{(2x)^4} \cdot \sqrt[4]{2x}$ . Because the index $4$ is even, absolute value is required: $\sqrt[4]{(2x)^4} = 2|x|$ . The simplified form is $2|x|\sqrt[4]{2x}$ .

Both examples illustrate the Quotient Property in reverse: when neither individual radicand is a perfect $n$ th power, combining them under a single radical can produce common factors that cancel, yielding a simpler radicand that can then be factored into a perfect $n$ th power times a remaining factor.

Updated 2026-05-01

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