If $1176 = 2^{a} \times 3^{b} \times 7^{c}$ , find the values of a, b and c. Hence, compute the value of $2^{a} \times 3^{b} \times 7^{- c}$ as a fraction.

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Solution

First find the prime factorisation of 1176.
$\begin{matrix} 2 & 1176 \\ 2 & 588 \\ 2 & 294 \\ 3 & 147 \\ 7 & 49 \\ 7 & 7 \\ 1 \end{matrix}$
It can be observed that 1176 can be written as $2^{3} \times 3^{1} \times 7^{2}$ .
$1176 = 2^{a} 3^{b} 7^{c} = 2^{3} 3^{1} 7^{2}$
So, a = 3, b = 1 and c = 2.
Therefore, the value of $2^{a} \times 3^{b} \times 7^{- c}$ is $2^{3} \times 3^{1} \times 7^{- 2} = 8 \times 3 \times \frac{1}{49} = \frac{24}{49}$

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