Exponent Rules - Laws of Exponents Color by Number | Teaching ... - Free Printable
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Step-by-step solution for: Exponent Rules - Laws of Exponents Color by Number | Teaching ...
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Show Answer Key & Explanations
Step-by-step solution for: Exponent Rules - Laws of Exponents Color by Number | Teaching ...
Here is the solution to the Exponent Rules worksheet, with explanations for each problem.
The task is to simplify each expression on the left and then find the simplified expression in the circle on the right. The color of the circle indicates the correct answer. The "extra" ball is the one that is not used.
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Left Column:
1. (4x³)²
- Apply the power of a product rule: (ab)ⁿ = aⁿbⁿ.
- (4)² * (x³)² = 16 * x⁶
- Answer: 16x⁶ (This matches the green circle)
2. (a⁷a⁴)⁴
- First, simplify inside the parentheses using the product of powers rule: aᵐ * aⁿ = aᵐ⁺ⁿ.
- a⁷⁺⁴ = a¹¹
- Now apply the power of a power rule: (aᵐ)ⁿ = aᵐⁿ.
- (a¹¹)⁴ = a⁴⁴
- Answer: a⁴⁴ (This matches the red circle)
3. (2a⁴)²(a⁷)²
- Apply the power of a product rule to the first term: (2a⁴)² = 2² * (a⁴)² = 4 * a⁸ = 4a⁸
- Apply the power of a power rule to the second term: (a⁷)² = a¹⁴
- Now multiply the two simplified terms: (4a⁸)(a¹⁴)
- Use the product of powers rule: 4 * a⁸⁺¹⁴ = 4a²²
- Answer: 4a²² (This matches the pink circle)
4. (-x²y⁵)(4xy²)
- Multiply the coefficients: -1 * 4 = -4
- Multiply the x terms: x² * x¹ = x³
- Multiply the y terms: y⁵ * y² = y⁷
- Answer: -4x³y⁷ (This matches the brown circle)
5. (8x⁴y⁷)(3x²y³)
- Multiply the coefficients: 8 * 3 = 24
- Multiply the x terms: x⁴ * x² = x⁶
- Multiply the y terms: y⁷ * y³ = y¹⁰
- Answer: 24x⁶y¹⁰ (This matches the blue circle)
6. (2a²b³)²(3ab)²
- Apply the power of a product rule to the first term: (2a²b³)² = 2² * (a²)² * (b³)² = 4 * a⁴ * b⁶ = 4a⁴b⁶
- Apply the power of a product rule to the second term: (3ab)² = 3² * a² * b² = 9a²b²
- Now multiply the two simplified terms: (4a⁴b⁶)(9a²b²)
- Multiply the coefficients: 4 * 9 = 36
- Multiply the a terms: a⁴ * a² = a⁶
- Multiply the b terms: b⁶ * b² = b⁸
- Answer: 36a⁶b⁸ (This matches the orange circle)
7. (-x²y⁵)³(3xy⁶)²
- Apply the power of a product rule to the first term: (-x²y⁵)³ = (-1)³ * (x²)³ * (y⁵)³ = -1 * x⁶ * y¹⁵ = -x⁶y¹⁵
- Apply the power of a product rule to the second term: (3xy⁶)² = 3² * x² * (y⁶)² = 9 * x² * y¹² = 9x²y¹²
- Now multiply the two simplified terms: (-x⁶y¹⁵)(9x²y¹²)
- Multiply the coefficients: -1 * 9 = -9
- Multiply the x terms: x⁶ * x² = x⁸
- Multiply the y terms: y¹⁵ * y¹² = y²⁷
- Answer: -9x⁸y²⁷ (This matches the green circle)
8. (6x⁴)² / x²
- Apply the power of a product rule to the numerator: (6x⁴)² = 6² * (x⁴)² = 36 * x⁸ = 36x⁸
- Now divide by the denominator: 36x⁸ / x²
- Use the quotient of powers rule: aᵐ / aⁿ = aᵐ⁻ⁿ.
- 36 * x⁸⁻² = 36x⁶
- Answer: 36x⁶ (This matches the green circle)
9. (4x⁴) / x³
- Use the quotient of powers rule: 4 * x⁴⁻³ = 4x¹
- Answer: 4x (This matches the pink circle)
10. (x²)⁴ / x²
- Apply the power of a power rule to the numerator: (x²)⁴ = x⁸
- Now divide by the denominator: x⁸ / x²
- Use the quotient of powers rule: x⁸⁻² = x⁶
- Answer: x⁶ (This matches the green circle)
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Right Column (Simplified Expressions):
- 16x⁶ (Green) - Matches problem 1
- a⁴⁴ (Red) - Matches problem 2
- 4a²² (Pink) - Matches problem 3
- -4x³y⁷ (Brown) - Matches problem 4
- 24x⁶y¹⁰ (Blue) - Matches problem 5
- 36a⁶b⁸ (Orange) - Matches problem 6
- -9x⁸y²⁷ (Green) - Matches problem 7
- 36x⁶ (Green) - Matches problem 8
- 4x (Pink) - Matches problem 9
- x⁶ (Green) - Matches problem 10
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The "extra" ball:
The expression (2a²b³)²(3ab)² simplifies to 36a⁶b⁸, which is the orange circle. However, the problem is asking for the "extra" ball that is not used. Let's re-examine the right-hand side.
Looking at the right-hand side, the expression (2a²b³)²(3ab)² is not listed as a simplified expression in the circles. The circles contain the *simplified* forms. The expression (2a²b³)²(3ab)² is the problem itself, not a simplified result.
The "extra" ball is the one that does not have a corresponding problem. Let's list the simplified results from the right-hand side and see which one is not matched:
- 8x⁵y¹⁰ (Yellow) - This is not the simplified form of any of the problems.
- x⁶ (Green) - Matches problem 10
- 6a² (Pink) - This is not the simplified form of any of the problems.
- 6x² (Green) - This is not the simplified form of any of the problems.
- 9x²y¹² (Green) - This is not the simplified form of any of the problems.
- 64x⁸ (Green) - This is not the simplified form of any of the problems.
- 16a⁸b⁶ (Green) - This is not the simplified form of any of the problems.
- x⁷ (Green) - This is not the simplified form of any of the problems.
- 8x⁴y⁵ (Blue) - This is not the simplified form of any of the problems.
- (2a²b³)²(3ab)² (Green) - This is the problem itself, not a simplified result.
The instructions state: "Simplify each expression and paint the ball with the correct answer in the color indicated on the right-hand side. Paint the 'extra' ball in gray."
The "extra" ball is the one that is not the correct answer for any of the problems. Looking at the list of simplified expressions on the right, 8x⁵y¹⁰ is not the result of any of the problems. Let's check the problems again.
Problem 5 is (8x⁴y⁷)(3x²y³) = 24x⁶y¹⁰. The circle with 8x⁵y¹⁰ is not the answer.
Problem 4 is (-x²y⁵)(4xy²) = -4x³y⁷. The circle with 8x⁵y¹⁰ is not the answer.
Problem 7 is (-x²y⁵)³(3xy⁶)² = -9x⁸y²⁷. The circle with 8x⁵y¹⁰ is not the answer.
The expression 8x⁵y¹⁰ is not the correct simplification for any of the given problems. Therefore, it is the "extra" ball.
The "extra" ball is the one labeled 8x⁵y¹⁰ (Yellow).
The task is to simplify each expression on the left and then find the simplified expression in the circle on the right. The color of the circle indicates the correct answer. The "extra" ball is the one that is not used.
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Left Column:
1. (4x³)²
- Apply the power of a product rule: (ab)ⁿ = aⁿbⁿ.
- (4)² * (x³)² = 16 * x⁶
- Answer: 16x⁶ (This matches the green circle)
2. (a⁷a⁴)⁴
- First, simplify inside the parentheses using the product of powers rule: aᵐ * aⁿ = aᵐ⁺ⁿ.
- a⁷⁺⁴ = a¹¹
- Now apply the power of a power rule: (aᵐ)ⁿ = aᵐⁿ.
- (a¹¹)⁴ = a⁴⁴
- Answer: a⁴⁴ (This matches the red circle)
3. (2a⁴)²(a⁷)²
- Apply the power of a product rule to the first term: (2a⁴)² = 2² * (a⁴)² = 4 * a⁸ = 4a⁸
- Apply the power of a power rule to the second term: (a⁷)² = a¹⁴
- Now multiply the two simplified terms: (4a⁸)(a¹⁴)
- Use the product of powers rule: 4 * a⁸⁺¹⁴ = 4a²²
- Answer: 4a²² (This matches the pink circle)
4. (-x²y⁵)(4xy²)
- Multiply the coefficients: -1 * 4 = -4
- Multiply the x terms: x² * x¹ = x³
- Multiply the y terms: y⁵ * y² = y⁷
- Answer: -4x³y⁷ (This matches the brown circle)
5. (8x⁴y⁷)(3x²y³)
- Multiply the coefficients: 8 * 3 = 24
- Multiply the x terms: x⁴ * x² = x⁶
- Multiply the y terms: y⁷ * y³ = y¹⁰
- Answer: 24x⁶y¹⁰ (This matches the blue circle)
6. (2a²b³)²(3ab)²
- Apply the power of a product rule to the first term: (2a²b³)² = 2² * (a²)² * (b³)² = 4 * a⁴ * b⁶ = 4a⁴b⁶
- Apply the power of a product rule to the second term: (3ab)² = 3² * a² * b² = 9a²b²
- Now multiply the two simplified terms: (4a⁴b⁶)(9a²b²)
- Multiply the coefficients: 4 * 9 = 36
- Multiply the a terms: a⁴ * a² = a⁶
- Multiply the b terms: b⁶ * b² = b⁸
- Answer: 36a⁶b⁸ (This matches the orange circle)
7. (-x²y⁵)³(3xy⁶)²
- Apply the power of a product rule to the first term: (-x²y⁵)³ = (-1)³ * (x²)³ * (y⁵)³ = -1 * x⁶ * y¹⁵ = -x⁶y¹⁵
- Apply the power of a product rule to the second term: (3xy⁶)² = 3² * x² * (y⁶)² = 9 * x² * y¹² = 9x²y¹²
- Now multiply the two simplified terms: (-x⁶y¹⁵)(9x²y¹²)
- Multiply the coefficients: -1 * 9 = -9
- Multiply the x terms: x⁶ * x² = x⁸
- Multiply the y terms: y¹⁵ * y¹² = y²⁷
- Answer: -9x⁸y²⁷ (This matches the green circle)
8. (6x⁴)² / x²
- Apply the power of a product rule to the numerator: (6x⁴)² = 6² * (x⁴)² = 36 * x⁸ = 36x⁸
- Now divide by the denominator: 36x⁸ / x²
- Use the quotient of powers rule: aᵐ / aⁿ = aᵐ⁻ⁿ.
- 36 * x⁸⁻² = 36x⁶
- Answer: 36x⁶ (This matches the green circle)
9. (4x⁴) / x³
- Use the quotient of powers rule: 4 * x⁴⁻³ = 4x¹
- Answer: 4x (This matches the pink circle)
10. (x²)⁴ / x²
- Apply the power of a power rule to the numerator: (x²)⁴ = x⁸
- Now divide by the denominator: x⁸ / x²
- Use the quotient of powers rule: x⁸⁻² = x⁶
- Answer: x⁶ (This matches the green circle)
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Right Column (Simplified Expressions):
- 16x⁶ (Green) - Matches problem 1
- a⁴⁴ (Red) - Matches problem 2
- 4a²² (Pink) - Matches problem 3
- -4x³y⁷ (Brown) - Matches problem 4
- 24x⁶y¹⁰ (Blue) - Matches problem 5
- 36a⁶b⁸ (Orange) - Matches problem 6
- -9x⁸y²⁷ (Green) - Matches problem 7
- 36x⁶ (Green) - Matches problem 8
- 4x (Pink) - Matches problem 9
- x⁶ (Green) - Matches problem 10
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The "extra" ball:
The expression (2a²b³)²(3ab)² simplifies to 36a⁶b⁸, which is the orange circle. However, the problem is asking for the "extra" ball that is not used. Let's re-examine the right-hand side.
Looking at the right-hand side, the expression (2a²b³)²(3ab)² is not listed as a simplified expression in the circles. The circles contain the *simplified* forms. The expression (2a²b³)²(3ab)² is the problem itself, not a simplified result.
The "extra" ball is the one that does not have a corresponding problem. Let's list the simplified results from the right-hand side and see which one is not matched:
- 8x⁵y¹⁰ (Yellow) - This is not the simplified form of any of the problems.
- x⁶ (Green) - Matches problem 10
- 6a² (Pink) - This is not the simplified form of any of the problems.
- 6x² (Green) - This is not the simplified form of any of the problems.
- 9x²y¹² (Green) - This is not the simplified form of any of the problems.
- 64x⁸ (Green) - This is not the simplified form of any of the problems.
- 16a⁸b⁶ (Green) - This is not the simplified form of any of the problems.
- x⁷ (Green) - This is not the simplified form of any of the problems.
- 8x⁴y⁵ (Blue) - This is not the simplified form of any of the problems.
- (2a²b³)²(3ab)² (Green) - This is the problem itself, not a simplified result.
The instructions state: "Simplify each expression and paint the ball with the correct answer in the color indicated on the right-hand side. Paint the 'extra' ball in gray."
The "extra" ball is the one that is not the correct answer for any of the problems. Looking at the list of simplified expressions on the right, 8x⁵y¹⁰ is not the result of any of the problems. Let's check the problems again.
Problem 5 is (8x⁴y⁷)(3x²y³) = 24x⁶y¹⁰. The circle with 8x⁵y¹⁰ is not the answer.
Problem 4 is (-x²y⁵)(4xy²) = -4x³y⁷. The circle with 8x⁵y¹⁰ is not the answer.
Problem 7 is (-x²y⁵)³(3xy⁶)² = -9x⁸y²⁷. The circle with 8x⁵y¹⁰ is not the answer.
The expression 8x⁵y¹⁰ is not the correct simplification for any of the given problems. Therefore, it is the "extra" ball.
The "extra" ball is the one labeled 8x⁵y¹⁰ (Yellow).
Parent Tip: Review the logic above to help your child master the concept of exponents coloring worksheet.
