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Select the equivalent expression.

[tex]\[ \left(\frac{3^{-6}}{7^{-3}}\right)^5 = ? \][/tex]

Choose one answer:

A. [tex]\(\frac{7^{15}}{3^{30}}\)[/tex]
B. [tex]\(\frac{3^{15}}{7^{30}}\)[/tex]
C. [tex]\(\frac{7^3}{3^{-30}}\)[/tex]


Sagot :

Let's analyze the given expression step-by-step:

The expression to simplify is:
[tex]\[ \left(\frac{3^{-6}}{7^{-3}}\right)^5 \][/tex]

First, let's simplify the expression inside the parentheses:
[tex]\[ \frac{3^{-6}}{7^{-3}} \][/tex]

Using the property of exponents [tex]\(\frac{a^m}{b^n} = a^m \cdot b^{-n}\)[/tex], we can rewrite:
[tex]\[ \frac{3^{-6}}{7^{-3}} = 3^{-6} \cdot 7^{3} \][/tex]

Now we raise this result to the power of 5:
[tex]\[ (3^{-6} \cdot 7^{3})^5 \][/tex]

Using the property [tex]\((a \cdot b)^m = a^m \cdot b^m\)[/tex], we split the powers:
[tex]\[ (3^{-6})^5 \cdot (7^{3})^5 \][/tex]

Now, simplify each part individually:
[tex]\[ (3^{-6})^5 = 3^{-6 \cdot 5} = 3^{-30} \][/tex]
[tex]\[ (7^{3})^5 = 7^{3 \cdot 5} = 7^{15} \][/tex]

Putting these simplified expressions back together, we have:
[tex]\[ 3^{-30} \cdot 7^{15} \][/tex]

This can be rewritten using a fraction to show the exponents clearly:
[tex]\[ \frac{7^{15}}{3^{30}} \][/tex]

Therefore, the equivalent expression is:
[tex]\[ \frac{7^{15}}{3^{30}} \][/tex]

So the correct answer is:
[tex]\[ \boxed{\frac{7^{15}}{3^{30}}} \][/tex]

This corresponds to option (A).