Question

Fill in the blanks:
(i) $\sqrt[3]{125\times 27}=3\times ...$
(ii) $\sqrt[3]{8\times ...}=8$
(iii) $\sqrt[3]{1728}=4\times ...$
(iv) $\sqrt[3]{480}=\sqrt[3]{3}\times 2\times \sqrt[3]{...}$
(v) $\sqrt[3]{}...=\sqrt[3]{7}\times \sqrt[3]{8}$
(vi) $\sqrt[3]{...}=\sqrt[3]{4}\times \sqrt[3]{5}\times \sqrt[3]{6}$
(vii) $\sqrt[3]{\frac{27}{125}}=\frac{...}{5}$
(viii) $\sqrt[3]{\frac{729}{1331}}=\frac{9}{...}$
(ix) $\sqrt[3]{\frac{512}{...}}=\frac{8}{13}$

Answer 1

(i) 5
$\sqrt[3]{125\times 27}=3\times \underline{\mathbf{5}}$

$\because$ $$\begin{gathered} \sqrt[3]{125\times 27}=\sqrt[3]{125}\times \sqrt[3]{27}=\sqrt[3]{5\times 5\times 5}\times \sqrt[3]{3\times 3\times 3} \\ =5\times 3 \end{gathered}$$
$=3\times 5$ (Commutative law)

(ii) $8\times 8=64$

$\because$ $\sqrt[3]{8\times \underline{\mathbf{8}\mathbf{\times }\mathbf{8}}}=8$

(iii) 3
$\because$ $\sqrt[3]{1728}=12=4\times 3$

(iv) 20

$\because$ $\sqrt[3]{480}=\sqrt[3]{\left\{2\times 2\times 2\right\}\times 2\times 2\times 3\times 5}=2\times \sqrt[3]{3}\times \sqrt[3]{5\times 2\times 2}=\sqrt[3]{3}\times 2\times \sqrt[3]{20}$

(v) $7\times 8=56$

$\because$ $\sqrt[3]{7\times 8}=\sqrt[3]{7}\times \sqrt[3]{8}$

(vi) $4\times 5\times 6=120$

$\because$ $\sqrt[3]{4\times 5\times 6}=\sqrt[3]{4}\times \sqrt[3]{5}\times \sqrt[3]{6}$

(vii) 3
$\because$ $\sqrt[3]{\frac{27}{125}}=\frac{\sqrt[3]{27}}{\sqrt[3]{125}}=\frac{3}{5}$

(viii) 11

$\because$ $\sqrt[3]{\frac{729}{1331}}=\frac{\sqrt[3]{729}}{\sqrt[3]{1331}}=\frac{9}{11}$

(ix) $13\times 13\times 13=2197$

$\because$ $\sqrt[3]{\frac{512}{{13}^3}}=\frac{\sqrt[3]{8^3}}{\sqrt[3]{{13}^3}}=\frac{8}{13}$