Sure, let's solve the given expression step-by-step.
We start with the expression:
[tex]\[ 3x y^3 \left(\frac{2}{5} x^4 y^3 + 3 x^3 y\right) \][/tex]
First, distribute [tex]\( 3x y^3 \)[/tex] across the terms inside the parentheses:
[tex]\[ 3x y^3 \cdot \left(\frac{2}{5} x^4 y^3\right) + 3x y^3 \cdot \left(3 x^3 y\right) \][/tex]
Now, let's handle each term separately.
For the first term:
[tex]\[ 3x y^3 \cdot \frac{2}{5} x^4 y^3 \][/tex]
Multiply the coefficients:
[tex]\[ 3 \cdot \frac{2}{5} = \frac{6}{5} \][/tex]
Simplify the powers:
[tex]\[ x \cdot x^4 = x^{1+4} = x^5 \][/tex]
[tex]\[ y^3 \cdot y^3 = y^{3+3} = y^6 \][/tex]
Thus, the first term simplifies to:
[tex]\[ \frac{6}{5} x^5 y^6 \][/tex]
For the second term:
[tex]\[ 3x y^3 \cdot 3 x^3 y \][/tex]
Multiply the coefficients:
[tex]\[ 3 \cdot 3 = 9 \][/tex]
Simplify the powers:
[tex]\[ x \cdot x^3 = x^{1+3} = x^4 \][/tex]
[tex]\[ y^3 \cdot y = y^{3+1} = y^4 \][/tex]
Thus, the second term simplifies to:
[tex]\[ 9 x^4 y^4 \][/tex]
Now combine the simplified terms:
[tex]\[ \frac{6}{5} x^5 y^6 + 9 x^4 y^4 \][/tex]
For simplicity, we multiply the first term by 5 to introduce a common denominator and rewrite the expression for clarity:
[tex]\[ \frac{6}{5} x^5 y^6 = \frac{6}{5} x^5 y^6 \cdot \frac{5}{5} = \frac{6 \cdot 5}{5 \cdot 5} x^5 y^6 = \frac{30}{25} x^5 y^6 \][/tex]
The final form of the expression, combining both simplified terms, is:
[tex]\[ x^4 y^4 \left(\frac{30}{25} x y^2 + 9 \right) \][/tex]
Or more clearly:
[tex]\[ x^4 y^4 (1.2 x y^2 + 9) \][/tex]
Thus, the simplified expression is:
[tex]\[ 3x y^3 \left(\frac{2}{5} x^4 y^3 + 3 x^3 y\right) = x^4 y^4 (1.2 x y^2 + 9) \][/tex]
