Let’s solve each problem step by step, simplifying using the
Laws of Exponents and writing answers with
only positive exponents.
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🔹 Left Column
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1.
Problem: \( 2x^2 \times 3x^3 \times 2x \)
- Multiply coefficients: \( 2 \times 3 \times 2 = 12 \)
- Add exponents of \( x \): \( x^2 \cdot x^3 \cdot x^1 = x^{2+3+1} = x^6 \)
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✔ Answer: \( \boxed{12x^6} \)
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2.
Problem: \( 4 \times 2x \times 3x^2y \)
- Coefficients: \( 4 \times 2 \times 3 = 24 \)
- Variables: \( x^1 \cdot x^2 = x^3 \), and \( y^1 \)
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✔ Answer: \( \boxed{24x^3y} \)
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3.
Problem: \( 4x \times x \times x^2 \)
- Coefficients: just 4
- Variables: \( x^1 \cdot x^1 \cdot x^2 = x^{1+1+2} = x^4 \)
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✔ Answer: \( \boxed{4x^4} \)
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4.
Problem: \( (2x^2)^3 \div 4x^2 \)
- First, simplify numerator: \( (2x^2)^3 = 2^3 \cdot (x^2)^3 = 8x^6 \)
- Now divide: \( \frac{8x^6}{4x^2} = \frac{8}{4} \cdot x^{6-2} = 2x^4 \)
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✔ Answer: \( \boxed{2x^4} \)
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5.
Problem: \( x^8 \div (x^3)^2 \)
- Simplify denominator: \( (x^3)^2 = x^{3 \cdot 2} = x^6 \)
- Divide: \( \frac{x^8}{x^6} = x^{8-6} = x^2 \)
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✔ Answer: \( \boxed{x^2} \)
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🔹 Right Column
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6.
Problem: \( 7x^8y^2 \div (x^3y)^2 \)
- Simplify denominator: \( (x^3y)^2 = (x^3)^2 \cdot y^2 = x^6y^2 \)
- Now divide: \( \frac{7x^8y^2}{x^6y^2} = 7 \cdot x^{8-6} \cdot y^{2-2} = 7x^2y^0 = 7x^2 \cdot 1 = 7x^2 \)
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✔ Answer: \( \boxed{7x^2} \)
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7.
Problem: \( \frac{4x^2 \times 3x^4}{6x^4} \)
- Numerator: \( 4 \times 3 = 12 \), \( x^2 \cdot x^4 = x^6 \) → so numerator is \( 12x^6 \)
- Denominator: \( 6x^4 \)
- Divide: \( \frac{12x^6}{6x^4} = 2x^{6-4} = 2x^2 \)
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✔ Answer: \( \boxed{2x^2} \)
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8.
Problem: \( \left( \frac{x^4}{y^2} \right)^3 \)
- Apply exponent to numerator and denominator:
\( \frac{(x^4)^3}{(y^2)^3} = \frac{x^{12}}{y^6} \)
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✔ Answer: \( \boxed{\frac{x^{12}}{y^6}} \)
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9.
Problem: \( \frac{x^8 \times (xy^2)^4}{(2x^2)^4} \)
- Step 1: Expand numerator:
- \( (xy^2)^4 = x^4(y^2)^4 = x^4y^8 \)
- So numerator: \( x^8 \cdot x^4y^8 = x^{12}y^8 \)
- Step 2: Expand denominator:
- \( (2x^2)^4 = 2^4 \cdot (x^2)^4 = 16x^8 \)
- Step 3: Divide:
- \( \frac{x^{12}y^8}{16x^8} = \frac{1}{16} \cdot x^{12-8} \cdot y^8 = \frac{x^4y^8}{16} \)
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✔ Answer: \( \boxed{\frac{x^4y^8}{16}} \)
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##
✔ Final Answers:
Left Column:
1. \( \boxed{12x^6} \)
2. \( \boxed{24x^3y} \)
3. \( \boxed{4x^4} \)
4. \( \boxed{2x^4} \)
5. \( \boxed{x^2} \)
Right Column:
6. \( \boxed{7x^2} \)
7. \( \boxed{2x^2} \)
8. \( \boxed{\frac{x^{12}}{y^6}} \)
9. \( \boxed{\frac{x^4y^8}{16}} \)
All answers use only
positive exponents, as required.
Parent Tip: Review the logic above to help your child master the concept of exponent laws worksheet.
