Sure, let's work through the problem step by step to simplify the expression:
Given expression:
[tex]\[ 15 a b^2 \times 3 a^2 b \div 5 a b^3 \][/tex]
First, let's rewrite the division as a multiplication by the reciprocal:
[tex]\[ 15 a b^2 \times 3 a^2 b \times \frac{1}{5 a b^3} \][/tex]
Now, combine the terms in the numerator:
[tex]\[ \frac{(15 a b^2 \times 3 a^2 b)}{5 a b^3} = \frac{15 \cdot 3 \cdot a \cdot a^2 \cdot b^2 \cdot b}{5 \cdot a \cdot b^3} = \frac{45 a^3 b^3}{5 a b^3} \][/tex]
Next, simplify the fraction. We can divide both the numerator and the denominator by [tex]\(5 a b^3\)[/tex]:
[tex]\[ \frac{45 a^3 b^3}{5 a b^3} = 9 \cdot \frac{a^3}{a} \cdot \frac{b^3}{b^3} \][/tex]
Notice that [tex]\( \frac{a^3}{a} \)[/tex] simplifies to [tex]\( a^2 \)[/tex], and [tex]\( \frac{b^3}{b^3} \)[/tex] simplifies to 1:
[tex]\[ 9 \cdot a^2 \][/tex]
Thus, the simplified expression is:
[tex]\[ 9a^2 \][/tex]
