Sure, let's carefully simplify the given expression and determine which option is correct.
We start with the given expression [tex]\(\left(x^2\right)^7\left(y^3\right)^5\)[/tex].
1. Simplify the first part: [tex]\(\left(x^2\right)^7\)[/tex]:
[tex]\[
\left(x^2\right)^7 = x^{2 \cdot 7} = x^{14}
\][/tex]
Here, we use the rule of exponents that states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
2. Simplify the second part: [tex]\(\left(y^3\right)^5\)[/tex]:
[tex]\[
\left(y^3\right)^5 = y^{3 \cdot 5} = y^{15}
\][/tex]
Similarly, we use the same rule of exponents.
So, the original expression simplifies to:
[tex]\[
x^{14} \cdot y^{15}
\][/tex]
Now, we look at the provided options to determine which one matches our simplified expression:
1. [tex]\(\left(x^2\right)^7\left(y^3\right)^5 = y^5 x^{14}\)[/tex]
[tex]\[
\text{This is not correct since the exponents do not match.}
\][/tex]
2. [tex]\(\left(x^2\right)^7\left(y^3\right)^5 = y^{14} x^{15}\)[/tex]
[tex]\[
\text{This is not correct since the exponents do not match.}
\][/tex]
3. [tex]\(\left(x^2\right)^7\left(y^3\right)^5 = y^{15} x^{14}\)[/tex]
[tex]\[
\text{This is correct since it matches } x^{14} \cdot y^{15}.
\][/tex]
4. [tex]\(\left(x^2\right)^7\left(y^3\right)^5 = y^{15} x^4\)[/tex]
[tex]\[
\text{This is not correct since the exponents do not match.}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
