1Cademy - Simplifying $$\sqrt[3]{\frac{a^8}{a^5}}$$ and $$\sqrt[4]{\frac{a^{10}}{a^2}}$$

Learn Before

Simplified Radical Expression

Example

Simplifying $\sqrt[3]{\frac{a^8}{a^5}}$ and $\sqrt[4]{\frac{a^{10}}{a^2}}$

Simplify two higher roots whose radicands are fractions of like bases by first reducing the fraction inside the radical using the Quotient Property for Exponents, then simplifying the resulting radical.

ⓐ $\sqrt[3]{\frac{a^8}{a^5}} = a$ :

Simplify the fraction under the radical first. Both the numerator and denominator share the base $a$ . Subtract the exponents: $\frac{a^8}{a^5} = a^{8-5} = a^3$ . The expression becomes $\sqrt[3]{a^3}$ . Since the index is $3$ and the radicand is a perfect cube, simplify: $\sqrt[3]{a^3} = a$ .

ⓑ $\sqrt[4]{\frac{a^{10}}{a^2}} = a^2$ :

Simplify the fraction under the radical first: $\frac{a^{10}}{a^2} = a^{10-2} = a^8$ . The expression becomes $\sqrt[4]{a^8}$ . Rewrite the radicand using perfect fourth power factors: $a^8 = (a^2)^4$ . Since $\sqrt[4]{(a^2)^4} = a^2$ , the simplified form is $a^2$ .

In both parts, the first step is to reduce the fraction inside the radical by subtracting exponents — the same technique used for simplifying square roots of quotients like $\sqrt{\frac{m^6}{m^4}}$ . After reducing, the resulting radicand may be a perfect $n$ th power that simplifies cleanly.

Updated 2026-05-01

Contributors are:

Who are from:

References

Learn Before

Related

Learn After