# Show that:

Question:

Show that:

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

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

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

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

Solution:

(i)

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

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

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

(ii)

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

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

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

(iii)

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

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

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

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

(iv)

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

$=\sqrt{\{-5 \times-5 \times-5\} \times\{-10 \times-10 \times-10\}}=-5 \times-10=50$

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

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