Sure! Let's solve each product step by step.
### 1. [tex]\((3x)\left(x^2 y^3\right)\)[/tex]
- Step 1: Identify the coefficients and variables in each term. Here, [tex]\(3x\)[/tex] and [tex]\(x^2 y^3\)[/tex].
- Step 2: Multiply the coefficients: [tex]\(3\)[/tex].
- Step 3: Apply the product rule for exponents: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex].
- For [tex]\(x\)[/tex]: [tex]\(x \cdot x^2 = x^{1+2} = x^3\)[/tex].
- For [tex]\(y\)[/tex]: Since [tex]\(y^3\)[/tex] does not have another [tex]\(y\)[/tex] term to multiply with, it stays as [tex]\(y^3\)[/tex].
Thus, the product is:
[tex]\[3x^3y^3\][/tex]
### 2. [tex]\(\left(5 a^3 b\right)(2 a b)\)[/tex]
- Step 1: Identify the coefficients and variables: [tex]\(5a^3b\)[/tex] and [tex]\(2ab\)[/tex].
- Step 2: Multiply the coefficients: [tex]\(5 \cdot 2 = 10\)[/tex].
- Step 3: Apply the product rule for exponents:
- For [tex]\(a\)[/tex]: [tex]\(a^3 \cdot a = a^{3+1} = a^4\)[/tex].
- For [tex]\(b\)[/tex]: [tex]\(b \cdot b = b^{1+1} = b^2\)[/tex].
Thus, the product is:
[tex]\[10a^4b^2\][/tex]
### 3. [tex]\(\left(18 y^2 x^3 z\right)\left(3 x^8 y^6 z^4\right)\)[/tex]
- Step 1: Identify the coefficients and variables: [tex]\(18y^2x^3z\)[/tex] and [tex]\(3x^8y^6z^4\)[/tex].
- Step 2: Multiply the coefficients: [tex]\(18 \cdot 3 = 54\)[/tex].
- Step 3: Apply the product rule for exponents:
- For [tex]\(y\)[/tex]: [tex]\(y^2 \cdot y^6 = y^{2+6} = y^8\)[/tex].
- For [tex]\(x\)[/tex]: [tex]\(x^3 \cdot x^8 = x^{3+8} = x^{11}\)[/tex].
- For [tex]\(z\)[/tex]: [tex]\(z \cdot z^4 = z^{1+4} = z^5\)[/tex].
Thus, the product is:
[tex]\[54y^8x^{11}z^5\][/tex]
### 4. [tex]\((2xyz)\left(-4x^2yz\right)\)[/tex]
- Step 1: Identify the coefficients and variables: [tex]\(2xyz\)[/tex] and [tex]\(-4x^2yz\)[/tex].
- Step 2: Multiply the coefficients: [tex]\(2 \cdot -4 = -8\)[/tex].
- Step 3: Apply the product rule for exponents:
- For [tex]\(x\)[/tex]: [tex]\(x \cdot x^2 = x^{1+2} = x^3\)[/tex].
- For [tex]\(y\)[/tex]: [tex]\(y \cdot y = y^{1+1} = y^2\)[/tex].
- For [tex]\(z\)[/tex]: [tex]\(z \cdot z = z^{1+1} = z^2\)[/tex].
Thus, the product is:
[tex]\[-8x^3y^2z^2\][/tex]
So, the final results for the products are:
1. [tex]\(3x^3y^3\)[/tex]
2. [tex]\(10a^4b^2\)[/tex]
3. [tex]\(54y^8x^{11}z^5\)[/tex]
4. [tex]\(-8x^3y^2z^2\)[/tex]
