1.9 Laws of indices
9 exam questions worth 21 marks. Write your answer, open the mark scheme, then give yourself the marks you earned. Your answers are saved on this device.
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Which of these is the simplified form of 3^4 \times 3^2?
Choose one answer.
Correct Not quite: the answer is \( 3^6 \)
Mark scheme
C: 3^6
Model answer
3^6 (C). Add the indices: 4 + 2 = 6.
Write each of these as a single power. (a) 7^5 \times 7^2 (b) 5^9 \div 5^4 (c) (2^3)^4
Mark scheme
B1: 7⁷; B1: 5⁵; B1: 2¹²
Model answer
(a) 7⁷ (b) 5⁵ (c) 2¹².
Simplify (a) 3x^2 \times 4x^5 (b) 20y^7 \div 4y^3 (c) (2p^3)^4.
Mark scheme
B1: 12x⁷; B1: 5y⁴; B1: 16p¹²
Model answer
(a) 3 × 4 = 12 and x² × x⁵ = x⁷, so 12x⁷. (b) 20 ÷ 4 = 5 and y⁷ ÷ y³ = y⁴, so 5y⁴. (c) 2⁴ = 16 and (p³)⁴ = p¹², so 16p¹².
Write down the value of (a) 6^0 (b) 2^{-3} (c) 10^{-2} as a decimal.
Mark scheme
B1: 1; B1: 1/8 (0.125); B1: 0.01
Model answer
(a) 1 (b) 1/8 (c) 0.01.
Work out \left(\dfrac{3}{4}\right)^{-2}.
Mark scheme
M1: (4/3)² (turning the fraction upside down); A1: 16/9
Model answer
A negative power turns the fraction upside down: (3/4)⁻² = (4/3)² = 16/9.
What is the value of 2^{-4}?
Choose one answer.
Correct Not quite: the answer is \( \frac{1}{16} \)
Mark scheme
C: \frac{1}{16}
Model answer
\frac{1}{16} (C). 2⁻⁴ = 1 / 2⁴ = 1/16.
Find the value of n in each case. (a) 2^n = 64 (b) 5^n = \frac{1}{25}
Mark scheme
B1: 6; B1: -2
Model answer
(a) 64 = 2⁶, so n = 6. (b) 1/25 = 5⁻², so n = -2.
Simplify (2a^2b)^3 \times 3ab^4.
Mark scheme
M1: (2a²b)³ = 8a⁶b³; M1: multiplies the numbers (24) and adds the powers; A1: 24a⁷b⁷
Model answer
(2a²b)³ = 8a⁶b³. Then 8a⁶b³ × 3ab⁴ = 24a⁷b⁷.
Show that \dfrac{2^5 \times 2^{-2}}{2^4} = \dfrac{1}{2}.
Mark scheme
M1: 2⁵ × 2⁻² = 2³; M1: 2³ ÷ 2⁴ = 2⁻¹; A1: 1/2
Model answer
On the top 2⁵ × 2⁻² = 2^(5 - 2) = 2³. Then 2³ ÷ 2⁴ = 2^(3 - 4) = 2⁻¹ = 1/2.
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