Answer:
[tex]\frac{7^{15}}{3^{30}}[/tex]
Step-by-step explanation:
The given expression is:
[tex](\frac{3^{-6}}{7^{-3}})^{5}[/tex]
Moving the expression to the other side in the fraction changes its sign to opposite. A numerator with negative exponent, when written in denominator will have the positive exponent. Using this rule, we can write:
[tex](\frac{3^{-6}}{7^{-3}})^{5}\\\\ = (\frac{7^{3}}{3^{6}} )^{5}[/tex]
The exponent 5 can be distributed to both numerator and denominator as shown:
[tex](\frac{7^{3}}{3^{6}} )^{5}\\\\ = \frac{(7^{3})^{5}}{(3^{6})^{5}}[/tex]
The power of a power can be written as a product. i.e.
[tex]\frac{(7^{3})^{5}}{(3^{6})^{5}}\\\\ =\frac{7^{15}}{3^{30}}[/tex]
So, the expression similar to the given expression and with positive exponents is: [tex]\frac{7^{15}}{3^{30}}[/tex]
The equivalent expression will be [tex]\dfrac{7^{15}}{3^{30}}[/tex]
Given the indices:
[tex](\dfrac{3^{-6}}{7^{-3}})^5 \\\\[/tex]
Using the law of indices which states:
[tex](a^m)^n = a^{mn}\\[/tex]
The expression becomes:
[tex]=\dfrac{3^{-6\times5}}{7^{-3\times 5}} \\= \frac{3^{-30}}{7^{-15}\\}\\= \frac{1}{3^{30}} \times 7^{15}\\= \frac{7^{15}}{3^{30}}[/tex]
Hence the equivalent expression will be [tex]\dfrac{7^{15}}{3^{30}}[/tex]
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