Question

If $2160=2^a{.3}^b{.5}^c$ Find $a,b,c$Hence, calculate the value of $3^a\times 2^{-b}\times 5^{-c}$

Answer 1

Finding the value of a, b and c
$2160=2^a{.3}^b{.5}^c$
∴ Prime factorization of
$2160=2\times 2\times 2\times 2\times 3\times 3\times 3\times 5$
$=2^4\times 3^3\times 5^5$

$\therefore 2^4\times 3^3\times 5^5=2^a{.3}^b{.5}^c$
By comparing the same bases and their indices, we get:
$a=4,b=3andc=1$

Calculating the value of $3^a\times 2^{-b}\times 5^{-c}$

$3^a\times 2^{-b}\times 5^{-c}$
$=3^4\times (2{)}^{-3}\times (5{)}^{-1}$
$=81\times \frac{1}{2^3}\times \frac{1}{5}$ $[\because a^{-1}=\frac{1}{a}]$
$=81\times \frac{1}{8\times 5}$
$=\frac{81}{40}$

Hence, $3^a\times 2^{-b}\times 5^{-c}$ is $\frac{81}{40}$.