Question

Find the cube roots of the following numbers.
(1) $8000$
(2) $729$
(3) $343$
(4) $-512$
(5) $-2744$
(6) $32768$

Answer 1

(1)

To find the cube root of $8000$, let us factorise $8000$ first.
$8000=2\times 2\times 2\times 2\times 2\times 2\times 5\times 5\times 5$

$=4\times 4\times 4\times 5\times 5\times 5$ $=4^3\times 5^3$

$=(4\times 5{)}^3$ [$\because a^m\times b^m=(a\times b{)}^m$]

$=(20{)}^3$

$\therefore ,\sqrt[3]{38000}=20$
(2)

To find the cube root of $729$, let us factorise $729$ first.
$729=3\times 3\times 3\times 3\times 3\times 3$

$=3^3\times 3^3$

$=9^3$

$\therefore ,\sqrt[3]{3729}=9$
(3)

To find the cube root of $343$, let us factorise $343$ first.
$343=7\times 7\times 7$

$=7^3$

$\therefore ,\sqrt[3]{3343}=7$
(4)

To find the cube root of $-512$, let us factorise $512$ first.
$512=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$

$=2^3\times 2^3\times 2^3$

$=(2\times 2\times 2{)}^3$

$=8^3$

$\therefore ,\sqrt[3]{3-512}=-8$
(5)

To find the cube root of $-2744$, let us factorise $2744$ first.
$2744=2\times 2\times 2\times 7\times 7\times 7$

$=2^3\times 7^3$

$=(2\times 7{)}^3$

$={14}^3$

$\therefore ,\sqrt[3]{3-2744}=-14$

(6)

To find the cube root of $32768$, let us factorise $32768$ first.
$32768=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$

$=2^3\times 2^3\times 2^3\times 2^3\times 2^3$

$=(2\times 2\times 2\times 2\times 2{)}^3$

$={32}^3$

$\therefore ,\sqrt[3]{332768}=32$