Iris' Math Power Quiz

Correct Answer: x⁸

Rationale: The product rule states that when multiplying like bases, you add the exponents (5 + 3 = 8).

Correct Answer: y⁶

Rationale: The quotient rule states that when dividing like bases, you subtract the exponents (8 - 2 = 6).

Correct Answer: a⁻²

Rationale: Using the product rule, you add the exponents: 2 + (-4) = -2.

Correct Answer: 1

Rationale: Any non-zero number divided by itself is 1. Also, using the quotient rule gives b⁽⁵⁻⁵⁾ = b⁰, which equals 1.

Correct Answer: 15x⁶

Rationale: Multiply the coefficients (5 × 3 = 15) and add the exponents of x (2 + 4 = 6).

Correct Answer: 3c⁴

Rationale: Divide the coefficients (12 ÷ 4 = 3) and subtract the exponents of c (7 - 3 = 4).

Correct Answer: z⁻⁵

Rationale: Using the product rule, you add the exponents: (-3) + (-2) = -5.

Correct Answer: m⁴

Rationale: Using the quotient rule, you subtract the exponents: 3 - (-1) = 3 + 1 = 4.

Correct Answer: a⁵b⁴

Rationale: Combine the like bases by adding their exponents: a⁽²⁺³⁾ and b⁽³⁺¹⁾.

Correct Answer: xy⁻²

Rationale: Apply the quotient rule to each base: x⁽⁴⁻³⁾ = x¹ and y⁽⁷⁻⁹⁾ = y⁻².

Correct Answer: x¹²

Rationale: The power of a power rule states that you multiply the exponents (4 × 3 = 12).

Correct Answer: 16y⁸

Rationale: The power of a product rule means the exponent 4 applies to both the 2 and the y². 2⁴ = 16 and (y²)⁴ = y⁸.

Correct Answer: a⁵ / b⁵

Rationale: The power of a quotient rule states the exponent applies to both the numerator and the denominator.

Correct Answer: c⁻¹⁰

Rationale: Using the power of a power rule, you multiply the exponents: (-2) × 5 = -10.

Correct Answer: x⁹y⁶

Rationale: The power of a product rule means the exponent 3 applies to x³ and y². Multiply the exponents for each: (x³)³ = x⁹ and (y²)³ = y⁶.

Correct Answer: m¹²/n⁸

Rationale: Apply the outer exponent 4 to both the numerator and denominator, using the power of a power rule: (m³)⁴ = m¹² and (n²)⁴ = n⁸.

Correct Answer: 1 / (9z²)

Rationale: The exponent -2 applies to both the 3 and the z: 3⁻²z⁻² = (1/3²) × (1/z²) = 1 / (9z²).

Correct Answer: -8a⁶

Rationale: The exponent 3 applies to -2 and a²: (-2)³ = -8 and (a²)³ = a⁶.

Correct Answer: 25/x²

Rationale: The negative exponent inverts the fraction to (5/x)². Then, apply the exponent 2: 5²/x² = 25/x².

Correct Answer: a⁶b⁻³

Rationale: Distribute the outer exponent 3 to both factors, multiplying exponents: (a²)³ = a⁶ and (b⁻¹)³ = b⁻³.

Correct Answer: 1

Rationale: The zero exponent rule states that any non-zero base raised to the power of 0 is 1.

Correct Answer: 1

Rationale: The exponent 0 applies to the entire non-zero base (3x²y), so the result is 1.

Correct Answer: -1

Rationale: Order of operations: The exponent 0 applies only to the 5, not the negative sign. So, this is -(5⁰) = -(1) = -1.

Correct Answer: 1

Rationale: The parentheses indicate that the exponent 0 applies to the entire base, which is -5. Any non-zero base to the 0 power is 1.

Correct Answer: 1/x

Rationale: The negative exponent rule states a⁻ⁿ = 1/aⁿ. So, x⁻¹ = 1/x¹.

Correct Answer: 1/16

Rationale: The negative exponent rule means 4⁻² = 1/4² = 1/16.

Correct Answer: b³

Rationale: A negative exponent in the denominator 'flips' to the numerator as a positive exponent: 1 / b⁻³ = b³.

Correct Answer: 5 / x⁴

Rationale: The exponent -4 applies only to x, so x⁻⁴ = 1/x⁴. The 5 stays in the numerator: 5 × (1/x⁴).

Correct Answer: 9/4

Rationale: First, the negative exponent flips the fraction: (3/2)². Then, square the top and bottom: 3²/2² = 9/4.

Correct Answer: 1

Rationale: Using the product rule: a³⁺⁽⁻³⁾ = a⁰. And a⁰ = 1. Alternatively, a⁻³ = 1/a³, so a³/a³ = 1.

Correct Answer: 4

Rationale: An exponent of 1/2 is the same as the square root. √16 = 4.

Correct Answer: 3

Rationale: An exponent of 1/3 is the same as the cube root. ³√27 = 3, because 3 × 3 × 3 = 27.

Correct Answer: 10

Rationale: An exponent of 1/2 means the square root. √100 = 10.

Correct Answer: 125

Rationale: The denominator 2 means square root (√25 = 5). The numerator 3 means cube (5³ = 125).

Correct Answer: 4

Rationale: The denominator 3 means cube root (³√8 = 2). The numerator 2 means square (2² = 4).

Correct Answer: 1/3

Rationale: The negative exponent means reciprocal: 1 / 81¹/⁴. The 1/4 power means the 4th root: ⁴√81 = 3. So, 1/3.

Correct Answer: 4

Rationale: The denominator 5 means 5th root (⁵√32 = 2). The numerator 2 means square (2² = 4).

Correct Answer: ⁴√x³

Rationale: The denominator (4) becomes the index of the root. The numerator (3) becomes the power of the base.

Correct Answer: y²/⁵

Rationale: The power (2) is the numerator of the fractional exponent, and the root index (5) is the denominator.

Correct Answer: 1/8

Rationale: Negative exponent: 1/4³/². Then, 4³/² is (√4)³ = 2³ = 8. So, 1/8.

Correct Answer: x¹y⁷ (or xy⁷)

Rationale: Combine like bases by adding exponents: x⁽² ⁺ ⁻¹⁾ = x¹ and y⁽³ ⁺ ⁴⁾ = y⁷.

Correct Answer: 3a³b⁻¹

Rationale: Divide coefficients: 15/5 = 3. Subtract exponents for a: 5-2 = 3. Subtract exponents for b: 2-3 = -1.

Correct Answer: 4x⁶y⁻²

Rationale: Distribute the outer exponent 2 to all factors: 2² = 4, (x³)² = x⁶, and (y⁻¹)² = y⁻².

Correct Answer: x⁶

Rationale: First, simplify inside the parentheses: x⁴/x² = x⁽⁴⁻²⁾ = x². Then, apply the outer exponent: (x²)³ = x⁶.

Correct Answer: a⁶b⁻⁹

Rationale: Apply the outer exponent -3 to both factors: (a⁻²)⁻³ = a⁶ and (b³)⁻³ = b⁻⁹.

Correct Answer: x²

Rationale: First, simplify the numerator: (x²)³ = x⁶. Then, solve the quotient: x⁶ / x⁴ = x⁽⁶⁻⁴⁾ = x².

Correct Answer: 9 / (m²n²)

Rationale: Simplify inside: 3m⁽²⁻³⁾n⁽¹⁻²⁾ = 3m⁻¹n⁻¹. Square: (3m⁻¹n⁻¹)² = 3²(m⁻¹)²(n⁻¹)² = 9m⁻²n⁻² = 9 / (m²n²).

Correct Answer: y⁵

Rationale: First, x⁰ = 1. The expression becomes y³ / y⁻². Using the quotient rule, y⁽³ ⁻ ⁻²⁾ = y⁽³⁺²⁾ = y⁵.

Correct Answer: 2x²

Rationale: Distribute the 1/4 exponent: 16¹/⁴ = 2 and (x⁸)¹/⁴ = x⁽⁸ × ¹/⁴⁾ = x².

Correct Answer: 3b / (2a²)

Rationale: Negative exponent flips: ( (27b³) / (8a⁶) )¹/³. Cube root top & bottom: (³√27 × ³√b³) / (³√8 × ³√a⁶) = 3b / (2a²).