Let’s go through each problem one by one. For each, we’ll:
1. Identify the mistake
2. Correct it
3. Justify (show) our reasoning
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a) 2⁵ = 10
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Mistake: They multiplied 2 × 5 instead of calculating 2 raised to the 5th power.
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Correct: 2⁵ = 32
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Justify: 2⁵ means 2 × 2 × 2 × 2 × 2 = 4 × 2 × 2 × 2 = 8 × 2 × 2 = 16 × 2 = 32
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b) (-2)³ = 8
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Mistake: They forgot that a negative number raised to an odd power stays negative.
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Correct: (-2)³ = -8
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Justify: (-2)³ = (-2) × (-2) × (-2) = 4 × (-2) = -8
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c) -6² = 36
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Mistake: The exponent applies only to 6, not to the negative sign. So it’s -(6²), not (-6)².
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Correct: -6² = -36
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Justify: Order of operations: exponents before negation. So 6² = 36, then apply the negative → -36
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d) x⁰ = 0
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Mistake: Any nonzero number (or variable) to the 0 power is 1, not 0.
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Correct: x⁰ = 1 (as long as x ≠ 0)
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Justify: This is a rule in math — anything (except 0) to the 0 power equals 1. Example: 5⁰ = 1, 100⁰ = 1
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e) x³ • x⁴ = x¹²
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Mistake: When multiplying powers with the same base, you add the exponents, not multiply them.
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Correct: x³ • x⁴ = x
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Justify: x³ • x⁴ = x^(3+4) = x⁷. Think: x•x•x • x•x•x•x = 7 x’s total.
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f) x¹⁰ / x⁵ = x²
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Mistake: When dividing powers with the same base, subtract the exponents, not divide them.
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Correct: x¹⁰ / x⁵ = x⁵
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Justify: x¹⁰ / x⁵ = x^(10-5) = x⁵. You’re canceling 5 x’s from top and bottom.
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g) (x³)⁵ = x⁸
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Mistake: When raising a power to another power, multiply the exponents, not add them.
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Correct: (x³)⁵ = x¹⁵
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Justify: (x³)⁵ means x³ • x³ • x³ • x³ • x³ = x^(3+3+3+3+3) = x¹⁵, or just 3×5=15.
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h) 7⁻² = -49
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Mistake: Negative exponent does NOT mean negative result. It means reciprocal.
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Correct: 7⁻² = 1/49
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Justify: a⁻ⁿ = 1/(aⁿ). So 7⁻² = 1/(7²) = 1/49
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i) (-3)⁴ = -81
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Mistake: A negative number raised to an even power becomes positive.
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Correct: (-3)⁴ = 81
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Justify: (-3)⁴ = (-3)×(-3)×(-3)×(-3) = 9 × 9 = 81. Two negatives make a positive, twice.
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j) x⁵ / x⁹ = x⁴
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Mistake: When dividing, subtract exponents: 5 - 9 = -4, not 4. Also, they may have done 9 - 5 by mistake.
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Correct: x⁵ / x⁹ = x⁻⁴ or 1/x⁴
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Justify: x⁵ / x = x^(5-9) = x⁻⁴ = 1/x⁴. You can’t have more x’s on top than bottom — so answer is fraction.
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Final Answer:
a) Mistake: Multiplied base and exponent. Correct: 2⁵ = 32
b) Mistake: Forgot negative stays for odd power. Correct: (-2)³ = -8
c) Mistake: Exponent doesn’t apply to negative sign. Correct: -6² = -36
d) Mistake: Anything to 0 power is 1, not 0. Correct: x⁰ = 1
e) Mistake: Added exponents wrong? No — should add, but they multiplied. Correct: x³•x⁴ = x⁷
f) Mistake: Divided exponents instead of subtracting. Correct: x¹⁰/x⁵ = x⁵
g) Mistake: Added exponents instead of multiplying. Correct: (x³)⁵ = x¹⁵
h) Mistake: Thought negative exponent means negative value. Correct: 7⁻² = 1/49
i) Mistake: Even power should be positive. Correct: (-3)⁴ = 81
j) Mistake: Subtracted backwards or ignored negative. Correct: x⁵/x⁹ = x⁴ or 1/x
Parent Tip: Review the logic above to help your child master the concept of exponent rules practice worksheet.
