Let's solve the given expression step-by-step: [tex]\(\frac{3^5 \times 25^5 \times 225}{9^3 \times 125^4}\)[/tex].
1. Simplify individual terms:
- [tex]\(3^5\)[/tex] stays the same.
- [tex]\(25^5\)[/tex] can be simplified. Note that 25 is [tex]\(5^2\)[/tex], so [tex]\(25^5 = (5^2)^5 = 5^{10}\)[/tex].
- [tex]\(225\)[/tex] can be simplified. Note that 225 is [tex]\(15^2\)[/tex], and since [tex]\(15 = 3 \times 5\)[/tex], we have [tex]\(225 = (3 \times 5)^2 = 3^2 \times 5^2\)[/tex].
- [tex]\(9^3\)[/tex] can be simplified. Note that 9 is [tex]\(3^2\)[/tex], so [tex]\(9^3 = (3^2)^3 = 3^6\)[/tex].
- [tex]\(125^4\)[/tex] can be simplified. Note that 125 is [tex]\(5^3\)[/tex], so [tex]\(125^4 = (5^3)^4 = 5^{12}\)[/tex].
2. Rewrite the expression with simplified terms:
[tex]\[
\frac{3^5 \times 5^{10} \times 3^2 \times 5^2}{3^6 \times 5^{12}}
\][/tex]
3. Combine like bases in the numerator:
- Combine powers of 3: [tex]\(3^5 \times 3^2 = 3^{5+2} = 3^7\)[/tex].
- Combine powers of 5: [tex]\(5^{10} \times 5^2 = 5^{10+2} = 5^{12}\)[/tex].
So, the expression now is:
[tex]\[
\frac{3^7 \times 5^{12}}{3^6 \times 5^{12}}
\][/tex]
4. Simplify the fraction by canceling common terms:
- For the powers of 3: [tex]\(\frac{3^7}{3^6} = 3^{7-6} = 3^1 = 3\)[/tex].
- For the powers of 5: [tex]\(\frac{5^{12}}{5^{12}} = 1\)[/tex].
After canceling the common terms, we are left with:
[tex]\[
3 \times 1 = 3
\][/tex]
Therefore, the simplified value of the given expression [tex]\(\frac{3^5 \times 25^5 \times 225}{9^3 \times 125^4}\)[/tex] is [tex]\(3\)[/tex].
