To solve the expression [tex]\(\left(\frac{(2.3)^5}{(0.9)^4}\right)^3\)[/tex], we need to apply the laws of exponents. Let me guide you through the steps:
1. Understand the Given Expression:
[tex]\[
\left(\frac{(2.3)^5}{(0.9)^4}\right)^3
\][/tex]
2. Apply the Power of a Quotient Rule:
The power of a quotient rule states that [tex]\(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\)[/tex]. So, we apply this to the given expression:
[tex]\[
\left(\frac{(2.3)^5}{(0.9)^4}\right)^3 = \frac{((2.3)^5)^3}{((0.9)^4)^3}
\][/tex]
3. Simplify the Exponents:
To simplify this further, use the power of a power rule, which states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
[tex]\[
((2.3)^5)^3 = (2.3)^{5 \cdot 3} = (2.3)^{15}
\][/tex]
[tex]\[
((0.9)^4)^3 = (0.9)^{4 \cdot 3} = (0.9)^{12}
\][/tex]
4. Combine the Results:
Now we combine these results back into our fraction:
[tex]\[
\frac{(2.3)^{15}}{(0.9)^{12}}
\][/tex]
So, the expression after applying the laws of exponents and simplifying becomes:
[tex]\[
\frac{(2.3)^{15}}{(0.9)^{12}}
\][/tex]
Thus, the correct multiple-choice answer is:
[tex]\[
\boxed{\frac{(2.3)^{15}}{(0.9)^{12}}}
\][/tex]
