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How do you simplify $\sqrt[3]{216, 000}$ $root3(216,000)$ ?

1 Answer

Jan 11, 2016

$\sqrt[3]{216, 000} = 60$ $root(3)(216,000) =60$

Explanation:

Extracting the most obvious cube:
$XXX 216, 000 = 216 \times 10^{3}$ $color(white)("XXX")216,000 = 216 xx 10^3$

Then factor out basic primes until we get to a complete factoring:
$XXX = 2 \times 108 \times 10^{3}$ $color(white)("XXX")=2xx108xx10^3$

$XXX = 2^{2} \times 54 \times 10^{3}$ $color(white)("XXX")=2^2xx54xx10^3$

$XXX = 2^{3} \times 27 \times 10^{3}$ $color(white)("XXX")=2^3xx27xx10^3$

$XXX = 2^{3} \times 3 \times 9 \times 10^{3}$ $color(white)("XXX")=2^3xx3xx9xx10^3$

$XXX = 2^{3} \times 3^{3} \times 10^{3}$ $color(white)("XXX")=2^3xx3^3xx10^3$
$XXXXXX$ $color(white)("XXXXXX")$ In practice we probably would have recognized the cube factors before going this far.

Therefore
$XXX \sqrt[3]{216, 000} = \sqrt[3]{2^{3} \times 3^{3} \times 10^{3}} = 2 \times 3 \times 10 = 60$ $color(white)("XXX")root(3)(216,000) = root(3)(2^3xx3^3xx10^3) = 2xx3xx10 =60$

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