Simplify expressions using the laws of exponents with positive exponents.

Let’s solve each problem step by step, using the laws of exponents. Remember:
- When multiplying like bases, add exponents: \( x^a \times x^b = x^{a+b} \)
- When dividing like bases, subtract exponents: \( x^a \div x^b = x^{a-b} \)
- When raising a power to another power, multiply exponents: \( (x^a)^b = x^{a \cdot b} \)
- Also, simplify coefficients (numbers) separately.

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Problem 1:
\( 2x^2 \times 3x^3 \times 2x = ? \)

Step 1: Multiply coefficients: \( 2 \times 3 \times 2 = 12 \)
Step 2: Add exponents of x: \( x^2 \times x^3 \times x^1 = x^{2+3+1} = x^6 \)
→ Final: \( 12x^6 \)

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Problem 2:
\( 7x^8y^2 \div (x^3y)^2 = ? \)

Step 1: Simplify denominator: \( (x^3y)^2 = x^{6}y^{2} \)
Step 2: Divide: \( \frac{7x^8y^2}{x^6y^2} = 7x^{8-6}y^{2-2} = 7x^2y^0 = 7x^2 \cdot 1 = 7x^2 \)
→ Final: \( 7x^2 \)

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Problem 3:
\( 4 \times 2x \times 3x^2y = ? \)

Step 1: Multiply coefficients: \( 4 \times 2 \times 3 = 24 \)
Step 2: Combine x terms: \( x \times x^2 = x^3 \), and keep y
→ Final: \( 24x^3y \)

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Problem 4:
\( \frac{4x^2 \times 3x^4}{6x^4} = ? \)

Step 1: Numerator: \( 4 \times 3 = 12 \), \( x^2 \times x^4 = x^6 \) → \( 12x^6 \)
Step 2: Divide by denominator: \( \frac{12x^6}{6x^4} = 2x^{6-4} = 2x^2 \)
→ Final: \( 2x^2 \)

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Problem 5:
\( 4x \times x \times x^2 = ? \)

Step 1: Coefficient: 4
Step 2: Exponents: \( x^1 \times x^1 \times x^2 = x^{1+1+2} = x^4 \)
→ Final: \( 4x^4 \)

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Problem 6:
\( \left( \frac{x^4}{y^2} \right)^3 = ? \)

Step 1: Apply exponent to numerator and denominator:
Numerator: \( (x^4)^3 = x^{12} \)
Denominator: \( (y^2)^3 = y^6 \)
→ Final: \( \frac{x^{12}}{y^6} \)

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Problem 7:
\( (2x^2)^3 \div 4x^2 = ? \)

Step 1: Expand numerator: \( (2x^2)^3 = 8x^6 \)
Step 2: Divide: \( \frac{8x^6}{4x^2} = 2x^{6-2} = 2x^4 \)
→ Final: \( 2x^4 \)

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Problem 8:
\( x^8 \div (x^3)^2 = ? \)

Step 1: Simplify denominator: \( (x^3)^2 = x^6 \)
Step 2: Divide: \( \frac{x^8}{x^6} = x^{8-6} = x^2 \)
→ Final: \( x^2 \)

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Problem 9:
\( \frac{x^8 \times (xy^2)^4}{(2x^2)^4} = ? \)

Step 1: Expand numerator:
\( (xy^2)^4 = x^4y^8 \)
So numerator: \( x^8 \times x^4y^8 = x^{12}y^8 \)

Step 2: Expand denominator:
\( (2x^2)^4 = 16x^8 \)

Step 3: Divide:
\( \frac{x^{12}y^8}{16x^8} = \frac{1}{16} x^{12-8} y^8 = \frac{1}{16}x^4y^8 \)

→ Final: \( \frac{x^4y^8}{16} \)

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Final Answer:
12x⁶, 7x², 24x³y, 2x², 4x⁴, x¹²/y⁶, 2x⁴, x², x⁴y⁸/16