Indices II (Copy)
Practice Questions — 2.4 Indices II
Question 1
Simplify:
(a) x³ × x⁵
(b) a⁷ ÷ a³
(c) (y⁴)³
Question 2
Simplify:
(a) (m²n³)²
(b) (2x³)⁴
(c) (3a²b³)³
Question 3
Simplify:
(a) x⁵ ÷ x⁻²
(b) a⁻³ × a⁷
(c) y⁻⁴ ÷ y²
Question 4
Simplify:
(a) (p³q⁻²)(p⁻¹q⁴)
(b) (x²y⁻³)²
(c) (m⁻²n³)³
Question 5
Simplify:
(a) (a²b³)/(ab²)
(b) (x⁴y⁻²)/(x²y³)
(c) (p⁻³q²)/(p²q⁻¹)
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 6
Simplify:
(a) (2x²y³)²
(b) (5a³b⁻²)³
(c) (x⁻²y⁴)²
Question 7
Simplify each to a single power:
(a) 3² × 3⁴
(b) 5⁷ ÷ 5³
(c) (7³)²
Question 8
Simplify:
(a) (2³)⁴
(b) (10⁻²)³
(c) (½)⁻³
Question 9
Simplify:
(a) (x½)²
(b) (y⅓)³
(c) (z¼)⁴
Question 10
Simplify:
(a) (a⅔)(a⅓)
(b) (b⁵/²) ÷ (b³/²)
(c) (c³/⁴)²
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 11
Solve for x:
(a) 2ˣ = 8
(b) 3ˣ = 27
(c) 5ˣ = 1/25
Question 12
Solve for x:
(a) 4ˣ = 64
(b) 9ˣ = 1/81
(c) 7ˣ = 49
Question 13
Solve for x:
(a) 2²ˣ = 32
(b) 3²ˣ = 81
(c) 5²ˣ = 125
Question 14
Simplify:
(a) (x²/³)(x⁴/³)
(b) (y⁵/⁶) ÷ (y¹/³)
(c) (z³/²)²
Question 15
Simplify and write in positive indices:
(a) (a⁻³b²)³
(b) (x²y⁻¹)⁻²
(c) (m⁻½n²)²
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 16
Simplify:
(a) (p²q⁻³)³
(b) (r⁻²s⁴)²
(c) (x³y⁻²z)⁻¹
Question 17
Solve for x:
(a) 2ˣ = 16
(b) 4ˣ = 1/16
(c) 9ˣ = 81
Question 18
Solve for x:
(a) 3²ˣ = 81
(b) 5ˣ⁺¹ = 125
(c) 2ˣ⁺² = 32
Question 19
Simplify:
(a) (x⁴y²)½
(b) (a³b⁶)⅓
(c) (m⁴n²)¼
Question 20
Simplify:
(a) (p⁶)⅓
(b) (q⁸)¼
(c) (r¹²)⅙
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Answer key and explanations — 2.4 Indices II
1.
(a) x³ × x⁵ = x⁸
(b) a⁷ ÷ a³ = a⁴
(c) (y⁴)³ = y¹²
Explanation: Use aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ.
2.
(a) (m²n³)² = m⁴n⁶
(b) (2x³)⁴ = 2⁴ × x¹² = 16x¹²
(c) (3a²b³)³ = 27a⁶b⁹
3.
(a) x⁵ ÷ x⁻² = x⁵⁻(⁻²) = x⁷
(b) a⁻³ × a⁷ = a⁴
(c) y⁻⁴ ÷ y² = y⁻⁶ = 1/y⁶
Explanation: Negative indices mean reciprocals.
4.
(a) (p³q⁻²)(p⁻¹q⁴) = p²q²
(b) (x²y⁻³)² = x⁴y⁻⁶ = x⁴ / y⁶
(c) (m⁻²n³)³ = m⁻⁶n⁹ = n⁹ / m⁶
5.
(a) (a²b³)/(ab²) = ab
(b) (x⁴y⁻²)/(x²y³) = x² / y⁵
(c) (p⁻³q²)/(p²q⁻¹) = p⁻⁵q³ = q³ / p⁵
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
6.
(a) (2x²y³)² = 4x⁴y⁶
(b) (5a³b⁻²)³ = 125a⁹b⁻⁶ = 125a⁹ / b⁶
(c) (x⁻²y⁴)² = x⁻⁴y⁸ = y⁸ / x⁴
7.
(a) 3² × 3⁴ = 3⁶ = 729
(b) 5⁷ ÷ 5³ = 5⁴ = 625
(c) (7³)² = 7⁶ = 117 649
8.
(a) (2³)⁴ = 2¹² = 4096
(b) (10⁻²)³ = 10⁻⁶ = 1/1 000 000
(c) (½)⁻³ = 2³ = 8
9.
(a) (x½)² = x¹ = x
(b) (y⅓)³ = y
(c) (z¼)⁴ = z
10.
(a) (a⅔)(a⅓) = a¹ = a
(b) (b⁵/²) ÷ (b³/²) = b¹ = b
(c) (c³/⁴)² = c⁶/⁴ = c³/²
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
11. Solve for x
(a) 2ˣ = 8 → 2³ → x = 3
(b) 3ˣ = 27 → 3³ → x = 3
(c) 5ˣ = 1/25 → 5⁻² → x = −2
12.
(a) 4ˣ = 64 → 4³ → x = 3
(b) 9ˣ = 1/81 → 9⁻² → x = −2
(c) 7ˣ = 49 → 7² → x = 2
13.
(a) 2²ˣ = 32 → 2²ˣ = 2⁵ → 2x = 5 → x = 2.5
(b) 3²ˣ = 81 → 3²ˣ = 3⁴ → 2x = 4 → x = 2
(c) 5²ˣ = 125 → 5²ˣ = 5³ → 2x = 3 → x = 1.5
14.
(a) (x²/³)(x⁴/³) = x⁶/³ = x²
(b) (y⁵/⁶) ÷ (y¹/³) = y⁵/⁶⁻¹/³ = y³/⁶ = y½
(c) (z³/²)² = z³
15.
(a) (a⁻³b²)³ = a⁻⁹b⁶ = b⁶ / a⁹
(b) (x²y⁻¹)⁻² = x⁻⁴y² = y² / x⁴
(c) (m⁻½n²)² = m⁻¹n⁴ = n⁴ / m
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
16.
(a) (p²q⁻³)³ = p⁶q⁻⁹ = p⁶ / q⁹
(b) (r⁻²s⁴)² = r⁻⁴s⁸ = s⁸ / r⁴
(c) (x³y⁻²z)⁻¹ = x⁻³y²z⁻¹ = y² / (x³z)
17. Solve for x
(a) 2ˣ = 16 → 2⁴ → x = 4
(b) 4ˣ = 1/16 → 4⁻² → x = −2
(c) 9ˣ = 81 → 9² → x = 2
18.
(a) 3²ˣ = 81 → 3²ˣ = 3⁴ → 2x = 4 → x = 2
(b) 5ˣ⁺¹ = 125 → 5ˣ⁺¹ = 5³ → x + 1 = 3 → x = 2
(c) 2ˣ⁺² = 32 → 2ˣ⁺² = 2⁵ → x + 2 = 5 → x = 3
19.
(a) (x⁴y²)½ = x²y¹
(b) (a³b⁶)⅓ = a¹b²
(c) (m⁴n²)¼ = m¹n½
20.
(a) (p⁶)⅓ = p²
(b) (q⁸)¼ = q²
(c) (r¹²)⅙ = r²
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course