Question

The value of $\frac{6^n\times 2^{2n}\times 3^{3n}}{{30}^n\times 3^{2n}\times 2^{3n}}$ is equal to:

• A. $1$
• B. ${\left(0.3\right)}^n$
• C. $3^{-n}$
• D. $3^{5n}$

Answer 1

Answer: (B)

${\left(0.3\right)}^n$

Consider $\frac{6^n\times 2^{2n}\times 3^{3n}}{{30}^n\times 3^{2n}\times 2^{3n}}$and simplify it as follows:

$\frac{6^n\times 2^{2n}\times 3^{3n}}{{30}^n\times 3^{2n}\times 2^{3n}}=\frac{2^n\times 3^n\times 2^{2n}\times 3^{3n}}{2^n\times 3^n\times 5^n\times 3^{2n}\times 2^{3n}}$

$=\frac{2^{2n+n}\times 3^{3n+n}}{2^{3n+n}\times 3^{2n+n}\times 5^n}=\frac{2^{3n}\times 3^{4n}}{2^{4n}\times 3^{3n}\times 5^n}$

$=\frac{3^{4n-3n}}{2^{4n-3n}\times 5^n}=\frac{3^n}{2^n\times 5^n}=\frac{3^n}{{10}^n}={\left(0.3\right)}^n$

Hence, $\frac{6^n\times 2^{2n}\times 3^{3n}}{{30}^n\times 3^{2n}\times 2^{3n}}={\left(0.3\right)}^n$.