To simplify [tex]\(3\left(2^2 \times 3^3 \times 5^4\right)^2\)[/tex], let's follow these steps:
1. Simplify the expression inside the parentheses:
[tex]\[
2^2 \times 3^3 \times 5^4
\][/tex]
2. Raise this entire expression to the power of 2:
[tex]\[
\left(2^2 \times 3^3 \times 5^4\right)^2
\][/tex]
When you raise a product of powers to another power, you multiply the exponents. Using the property [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[
(2^2)^2 \times (3^3)^2 \times (5^4)^2
\][/tex]
Simplify each term inside the parentheses:
[tex]\[
2^{2 \times 2} \times 3^{3 \times 2} \times 5^{4 \times 2}
\][/tex]
3. Calculate the exponents:
[tex]\[
2^4 \times 3^6 \times 5^8
\][/tex]
4. Now include the multiplication by 3:
[tex]\[
3 \times \left(2^4 \times 3^6 \times 5^8\right)
\][/tex]
We can rewrite this by distributing the multiplication:
[tex]\[
3^1 \times 2^4 \times 3^6 \times 5^8
\][/tex]
5. Combine the like bases:
[tex]\[
2^4 \times 3^{1+6} \times 5^8
\][/tex]
Simplify the exponents:
[tex]\[
2^4 \times 3^7 \times 5^8
\][/tex]
Therefore, the expression [tex]\(3\left(2^2 \times 3^3 \times 5^4\right)^2\)[/tex] simplifies to:
[tex]\[
2^4 \times 3^7 \times 5^8
\][/tex]
