Let's solve the given mathematical expression step-by-step:
[tex]\[
(25)^{7.5} \times (5)^{2.5} \div (125)^{1.5} = 5^x
\][/tex]
First, we need to express all terms with the same base. We know that:
[tex]\[
25 = 5^2 \quad \text{and} \quad 125 = 5^3
\][/tex]
So, we rewrite the equation using these equivalent expressions:
[tex]\[
(5^2)^{7.5} \times (5)^{2.5} \div (5^3)^{1.5}
\][/tex]
Next, we use the power of a power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to simplify the exponents:
[tex]\[
5^{2 \cdot 7.5} \times 5^{2.5} \div 5^{3 \cdot 1.5}
\][/tex]
Calculate the exponents:
[tex]\[
5^{15} \times 5^{2.5} \div 5^{4.5}
\][/tex]
Now, we combine the exponents of the terms with the same base. Recall the rule [tex]\(a^m \times a^n = a^{m+n}\)[/tex] and [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
5^{15 + 2.5} \div 5^{4.5}
\][/tex]
Simplify the exponents:
[tex]\[
5^{17.5} \div 5^{4.5}
\][/tex]
Now, subtract the exponents in the denominator from the exponent in the numerator:
[tex]\[
5^{17.5 - 4.5}
\][/tex]
Perform the subtraction:
[tex]\[
5^{13}
\][/tex]
Therefore, we have:
[tex]\[
5^x = 5^{13}
\][/tex]
This implies that:
[tex]\[
x = 13
\][/tex]
So, the value of [tex]\(x\)[/tex] is:
[tex]\[
\boxed{13}
\][/tex]
