Show that:
Question:

Show that:

(i) $\sqrt[3]{27} \times \sqrt[3]{64}=\sqrt[3]{27 \times 64}$

(ii) $\sqrt[3]{64 \times 729}=\sqrt[3]{64} \times \sqrt[3]{729}$

(iv) $\sqrt[3]{-125 \times 216}=\sqrt[3]{-125} \times \sqrt[3]{216}$

(v) $\sqrt[3]{-125-1000}=\sqrt[3]{-125} \times \sqrt[3]{-1000}$

Solution:

(i)

LHS $=\sqrt[3]{27} \times \sqrt[3]{64}=\sqrt[3]{3 \times 3 \times 3} \times \sqrt[3]{4 \times 4 \times 4}=3 \times 4=12$

RHS $=\sqrt[3]{27 \times 64}=\sqrt[3]{3 \times 3 \times 3 \times 4 \times 4 \times 4}=\sqrt[3]{\{3 \times 3 \times 3\} \times\{4 \times 4 \times 4\}}=3 \times 4=12$

Because LHS is equal to RHS, the equation is true.

(ii)

LHS $=\sqrt[3]{64 \times 729}=\sqrt[3]{4 \times 4 \times 4 \times 9 \times 9 \times 9}=\sqrt[3]{\{4 \times 4 \times 4\} \times\{9 \times 9 \times 9\}}=4 \times 9=36$

RHS $=\sqrt[3]{64} \times \sqrt[3]{729}=\sqrt[3]{4 \times 4 \times 4} \times \sqrt[3]{9 \times 9 \times 9}=4 \times 9=36$

Because LHS is equal to RHS, the equation is true.

(iii)

LHS $=\sqrt[3]{-125 \times 216}=\sqrt[3]{-5 \times-5 \times-5 \times\{2 \times 2 \times 2 \times 3 \times 3 \times 3\}}$

$=\sqrt[3]{\{-5 \times-5 \times-5\} \times\{2 \times 2 \times 2\} \times\{3 \times 3 \times 3\}}=-5 \times 2 \times 3=-30$

RHS $=\sqrt[3]{-125} \times \sqrt[3]{216}=\sqrt[3]{-5 \times-5 \times-5} \times \sqrt[3]{\{2 \times 2 \times 2\} \times\{3 \times 3 \times 3\}}=-5 \times(2 \times 3)=-30$

Because LHS is equal to RHS, the equation is true.

(iv)

LHS $=\sqrt[3]{-125 \times-1000}=\sqrt[3]{-5 \times-5 \times-5 \times-10 \times-10 \times-10}$

$=\sqrt[3]{\{-5 \times-5 \times-5\} \times\{-10 \times-10 \times-10\}}=-5 \times-10=50$

RHS $=\sqrt[3]{-125} \times \sqrt[3]{-1000}=\sqrt[3]{-5 \times-5 \times-5} \times \sqrt[3]{\{-10 \times-10 \times-10\}}=-5 \times-10=50$

Because LHS is equal to RHS, the equation is true.