1. The problem is to simplify the algebraic expression $3uv \times v^2 \times 5u^2$. 2. The formula for multiplying terms with the same base is $a^m \times a^n = a^{m+n}$. 3. First, group the coefficients and the variables separately: $$3 \times 5 \times u \times u^2 \times v^2 \times v$$ 4. Multiply the coefficients: $$3 \times 5 = 15$$ 5. Apply the exponent addition rule to the variables: $$u^{1+2} = u^3$$ $$v^{2+1} = v^3$$ 6. Combine all parts: $$15u^3v^3$$ 7. Therefore, the simplified expression is $15u^3v^3$.