The answer is 8
Here's why:
[tex] { ( \frac{( {6}^{7}) \times ( {3}^{3}) }{( {6}^{6}) \times ( {3}^{4} ) } )}^{3} = \\ ( \frac{6}{3} ) ^{3} = \\ \frac{216}{27} = 8[/tex]
The exponents are subtracted one from another when divided.
[tex] \frac{ a ^{b} }{ {a}^{c} } = {a}^{b - c} [/tex]
We can look at the problem this way:
[tex]( \frac{6^{7} }{6 ^{6} } \times \frac{3^{3} }{ {3}^{4} } ) = (6^{7 - 6} \times {3}^{3 - 4} ) = \\ ({6}^{1} \times {3}^{ - 1} )[/tex]
Since we have the power of -1 on the 3, we apply this rule:
[tex] {a}^{ - b} = \frac{1}{ {a}^{b} } [/tex]
Also this rule because we have the power of 1 on the 6:
[tex] {a}^{1} = a[/tex]
Then we get this:
[tex](6 \times \frac{1}{3} )^{3} = ( \frac{6}{3} )^{3} [/tex]
We apply the rule:
[tex]( \frac{a}{b}) ^{c} = \frac{ {a}^{c} }{ {b}^{c} } [/tex]
We get this:
[tex] \frac{{6}^{3} }{ {3}^{3} } = \frac{216}{27} = 8[/tex]
