Simplify \sqrt [3]{486 x^3}

Question:

Simplify {eq}\sqrt [3]{486 x^3} {/eq}

Simplifying radicals


To simplify an expression involving radicals, we will factor out the quantity under the radical in factors that are powers with exponents that can be simplified with the order of the radical.

For example, {eq}\displaystyle \sqrt[n]{a} , {/eq} can be simplified if we extract powers with exponent {eq}\displaystyle n: a=b^n\cdot c^n\cdot d\implies \sqrt[n]{a}=\sqrt[n]{b^n\cdot c^n\cdot d}=bc\sqrt[n]{d}. {/eq}

This is because {eq}\displaystyle \sqrt[n]{a}=a^{\frac{1}{n}}. {/eq}

Answer and Explanation:


To simplify {eq}\displaystyle \sqrt[3]{486x^3} {/eq}

we will extract cubes outside the radical symbol.

First {eq}\displaystyle 486=2\cdot 243=2\cdot 3\cdot 81=2\cdot 3\cdot 3^4=2\cdot 3^2\cdot 3^3\\ \implies \sqrt[3]{486x^3}= \sqrt[3]{2\cdot 3^2\cdot 3^3\cdot x^3}=3x\sqrt[3]{2\cdot 3^2}=\boxed{3x\sqrt[3]{18}}. {/eq}


Learn more about this topic:

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Multiplying then Simplifying Radical Expressions

from Algebra I: High School

Chapter 7 / Lesson 10
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