Question 512 Marks
Simplify: $\frac{12^{4} \times 9^{3} \times 4}{6^{3} \times 8^{2} \times 27}$
Answer
View full question & answer→We have$\frac{12^{4} \times 9^{3} \times 4}{6^{3} \times 8^{2} \times 27}$
$=\frac{\left(2^{2} \times 3\right)^{4} \times\left(3^{2}\right)^{3} \times 2^{2}}{(2 \times 3)^{3} \times\left(2^{3}\right)^{2} \times 3^{3}}$
$=\frac{\left(2^{2}\right)^{4} \times(3)^{4} \times 3^{2 \times 3} \times 2^{2}}{2^{3} \times 3^{3} \times 2^{2 \times 3} \times 3^{3}}$
$=\frac{2^{8} \times 2^{2} \times 3^{4} \times 3^{6}}{2^{3} \times 2^{6} \times 3^{3} \times 3^{3}}$
$=\frac{2^{8+2} \times 3^{4+6}}{2^{3+6} \times 3^{3+3}}=\frac{2^{10} \times 3^{10}}{2^{9} \times 3^{6}}$
$=2^{10-9} \times 3^{10-6}=2^{1} \times 3^{4}$
$= 2 \times 81 = 162$
$=\frac{\left(2^{2} \times 3\right)^{4} \times\left(3^{2}\right)^{3} \times 2^{2}}{(2 \times 3)^{3} \times\left(2^{3}\right)^{2} \times 3^{3}}$
$=\frac{\left(2^{2}\right)^{4} \times(3)^{4} \times 3^{2 \times 3} \times 2^{2}}{2^{3} \times 3^{3} \times 2^{2 \times 3} \times 3^{3}}$
$=\frac{2^{8} \times 2^{2} \times 3^{4} \times 3^{6}}{2^{3} \times 2^{6} \times 3^{3} \times 3^{3}}$
$=\frac{2^{8+2} \times 3^{4+6}}{2^{3+6} \times 3^{3+3}}=\frac{2^{10} \times 3^{10}}{2^{9} \times 3^{6}}$
$=2^{10-9} \times 3^{10-6}=2^{1} \times 3^{4}$
$= 2 \times 81 = 162$