Simplify:
6x4 × 6x3
6x4 × 6x3
Worked Solution & Answer
Multiply the numbers and add the powers of matching letters.
e.g. 3b2 × 4b5 = 12b7
36x7
e.g. 3b2 × 4b5 = 12b7
36x7
Answer: 36x7
Indices (index laws) — Notes & Worked Examples Show / Hide
Explanation:
Learn the four core index laws for powers with the same base: (1) Multiply → add powers: am × an = am+n; (2) Divide → subtract powers: am ÷ an = am−n; (3) Brackets → multiply powers: (am)n = amn; (4) Power of zero: a0 = 1.
When an expression includes normal numbers (coefficients) in front of the letters, treat the numbers as normal numbers (multiply, divide, or raise them to the power) and apply the index laws to the powers!
Example 1: Simplify:
(a) y7 × y4y3 (b) (2x3y5)4
Solution:
(a) Top: y7+4 = y11; then y11−3 = y8
(b) Raise 2 to power 4: 24 = 16
Multiply powers: x3×4y5×4 = x12y20
Answer: (a) y8 (b) 16x12y20
Example 2: Simplify 18a6b23a2b5 and find the value of n in 5n × 53 = 511.
Solution:
Numbers: 18 ÷ 3 = 6; Powers: a6−2 = a4, b2−5 = b−3
For 5n × 53 = 511: n + 3 = 11 → n = 8
Answer: 6a4b−3 (or 6a4b3) and n = 8
Simplify:
20y95y5
20y95y5
Worked Solution & Answer
When dividing with the same base, subtract the powers.
e.g. b7 ÷ b3 = b4
4y4
e.g. b7 ÷ b3 = b4
4y4
Answer: 4y4
@@TABULAR0@@
Match each expression on the left to the correct expression on the right.
One has been done for you.
Match each expression on the left to the correct expression on the right.
One has been done for you.
Given 7ⁿ = x, match 7⁻ⁿ, 7²ⁿ, 7ⁿ⁺¹, 7 {}1/2 n) to 7x, 1/x, 1/2 x, √x, x². [3 marks]
Worked Solution & Answer
7²ⁿ = (7ⁿ)² = x². M1
7ⁿ⁺¹ = 7¹ × 7ⁿ = 7x. M1
7 {}1/2 n) = (7ⁿ) {}1/2) = √x. A1
7ⁿ⁺¹ = 7¹ × 7ⁿ = 7x. M1
7 {}1/2 n) = (7ⁿ) {}1/2) = √x. A1
Answer: 7²ⁿ → x², 7ⁿ⁺¹ → 7x, 7^(1/2 n) → √x
Write 27 × (32)7 as a single power of 3
Write your answer in the form 3n
Answer
Worked Solution & Answer
27 = 33. M1
(32)7 = 3^{2 × 7} = 3^{14}. M1
27 × (32)7 = 33 × 3^{14} = 3^{3 + 14} = 3^{17}. A1
(32)7 = 3^{2 × 7} = 3^{14}. M1
27 × (32)7 = 33 × 3^{14} = 3^{3 + 14} = 3^{17}. A1
Answer: Answer 317
(a) Simplify fully 2x³y⁵ × 7x²y (2)
(b) Simplify (m²)⁻³ (1)
(b) Simplify (m²)⁻³ (1)
(a) Simplify fully 2x3y5 × 7x2y [2 marks]
(b) Simplify (m2)^{-3} [1 mark]
Worked Solution & Answer
(a)
2×7 = 14, x³⁺² = x⁵, y⁵⁺¹ = y⁶ M1 A1
Answer (a): 14x5y6
(b)
m²×⁻³ = m⁻⁶ B1
Answer (b): m-6
(a) Simplify (p²)⁵ (1) (b) Simplify 12x⁷y³ ÷ 6x³y (2)
(a) Simplify (p2)5 [1 mark]
(b) Simplify 12x7y3 ÷ 6x3y [2 marks]
Worked Solution & Answer
(a)
p {}2×5) = p¹⁰ B1
Answer (a): p10
(b)
(12÷6) × x⁷⁻³ × y³⁻¹ = 2x⁴y² M1 A1
Answer (b): 2x4y2
Work out
(a) Simplify.
2 × 3a
(b) Simplify.
2x54x
(a) Simplify.
2 × 3a
(b) Simplify.
2x54x
(a) Simplify.
2 × 3a [1 mark]
2 × 3a [1 mark]
(b) Simplify.
2x54x [2 marks]
2x54x [2 marks]
Worked Solution & Answer
(a)
2 × 3a = 6a. B1
Answer (a): 6a
(b)
2/4 = 1/2 and x⁵ ÷ x = x⁴ M1
⇒ x⁴ / 2 (or 0.5x⁴). A1
⇒ x⁴ / 2 (or 0.5x⁴). A1
Answer (b): x42
(a) 2⁷ × 4⁵ = 4ˣ, find x (2) (b) Simplify fully (125p⁶y²⁴) {}2/3) (2)
(a) Calculate the value of x [2 marks]
x =
(b) Simplify fully (125p6y^{24})^{23} [2 marks]
Worked Solution & Answer
(a)
2⁷ × 2¹⁰ = 2²ˣ M1
⇒ 2x = 17
⇒ x = 8.5 A1
⇒ 2x = 17
⇒ x = 8.5 A1
Answer (a): x = 8.5
(b)
125 {}2/3) p {}6×2/3) y {}24×2/3) = 25p⁴y¹⁶ M1 A1
Answer (b): 25p4y^{16}
(a) x⁵ × x⁷ = xᵐ, find m. (1) (b) y⁸ ÷ y³ = yⁿ, find n. (1) (c) Simplify fully (5a⁴r²)³. (2)
(a) Find the value of m [1 mark]
m =
(b) Find the value of n [1 mark]
n =
(c) Simplify fully (5a4r2)3 [2 marks]
Worked Solution & Answer
(a)
5 + 7 = 12 B1
Answer (a): m = 12
(b)
8 – 3 = 5 B1
Answer (b): n = 5
(c)
5³ a {}4×3) r {}2×3) = 125a¹²r⁶ M1 A1
Answer (c): 125a12r6
(a) Factorise x2 - 5x - 24.
(b) Simplify (3h2)3.
(c) Represent the inequality -2 < x ≤ 3 on the number line below.
(b) Simplify (3h2)3.
(c) Represent the inequality -2 < x ≤ 3 on the number line below.
(a) Factorise x2 - 5x - 24. [2 marks]
(b) Simplify (3h2)3. [2 marks]
(c) Represent the inequality -2 < x ≤ 3 on the number line below. [1 mark]
Worked Solution & Answer
(a)
Two numbers that multiply to -24 and add to -5 are -8 and 3: (x - 8)(x + 3). M1 A1
Answer (a): (x - 8)(x + 3)
(b)
33 × (h2)3 = 27h^{2 × 3} = 27h6. M1 A1
Answer (b): 27h6
(c)
< requires an empty circle at -2
≤ requires a solid circle at 3
line connects them. B1
≤ requires a solid circle at 3
line connects them. B1
Answer (c): Open circle at -2, filled circle at 3, connected with a line