Sure! Let's break down the steps to simplify the given expression [tex]\(2 \sqrt{5x^3} \left( -3 \sqrt{10x^2} \right)\)[/tex]:
### Step 1: Multiply the Coefficients
First, multiply the coefficients outside of the radicals:
[tex]\[ 2 \times (-3) = -6 \][/tex]
### Step 2: Multiply the Radicands
Next, multiply the radicands under the square roots:
[tex]\[ (5x^3) \times (10x^2) = 5 \times 10 \times x^3 \times x^2 \][/tex]
Calculating the product of the numeric values:
[tex]\[ 5 \times 10 = 50 \][/tex]
Combining the variables with similar bases:
[tex]\[ x^3 \times x^2 = x^{3+2} = x^5 \][/tex]
So, the product of the radicands is:
[tex]\[ 50x^5 \][/tex]
### Step 3: Simplify
Combining the results of these steps, the expression now looks like:
[tex]\[ -6 \sqrt{50 x^5} \][/tex]
Thus, the simplified product is:
[tex]\[ -6 \sqrt{50 x^5} \][/tex]
There you have it! The product [tex]\(( 2 \sqrt{5 x^3}) \left( -3 \sqrt{10 x^2}) \)[/tex] simplifies down to:
[tex]\[ -6 \sqrt{50 x^5} \][/tex]
