Fractional and Negative Indices - Worksheets, Questions and Revision

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6.2 Fractional and Negative Indices

EDEXCEL 1MA1 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write down the value of:

(a)6⁰(1)

(b)2^-3(1)

(Total for Question 1 is 3 marks)

Which of the following is equal to 5^-2?

A) 25
B) -25
C) 1/25
D) -1/25

(Total for Question 2 is 1 mark)

Work out the value of:

(a)64^1/2(1)

(b)8^1/3(1)

(Total for Question 3 is 3 marks)

Work out the value of:

(a)4^-1/2(2)

(b)9^-3/2(2)

(Total for Question 4 is 4 marks)

Simplify, leaving your answer in index form:

(a)x⁵ x x^-2(1)

(b)y³ / y^-4(1)

(c)(m^-2)³(1)

(Total for Question 5 is 3 marks)

Which of the following is equal to 1/(x³)?

A) x^-3
B) x³
C) -x³
D) x^1/3

(Total for Question 6 is 1 mark)

Work out the value of:

(a)8^2/3(2)

(b)25^3/2(2)

(Total for Question 7 is 6 marks)

Simplify fully:

(a)(x⁴ y^-2) / (x^-1 y³)(2)

(b)Write (2a³)^-2 in the form k aⁿ, stating the values of k and n.(2)

(Total for Question 8 is 4 marks)

Solve the equation 2^x = 1/32

(Total for Question 9 is 2 marks)

Solve the equations:

(a)x^1/2 = 7(1)

(b)x^2/3 = 25(2)

(Total for Question 10 is 3 marks)

Show that 9^-1/2 x 3⁴ = 27

(Total for Question 11 is 3 marks)

Work out the value of 27^-2/3. Give your answer as a fraction.

(Total for Question 12 is 3 marks)

The number of bacteria, P, in a culture is modelled by P = 200 x 3^t, where t is the time in hours after the culture begins.

(a)Work out the number of bacteria when t = 0.(1)

(b)Work out the number of bacteria after 4 hours.(2)

(Total for Question 13 is 3 marks)

The mass, M grams, of a radioactive sample after t years is given by M = 80 x 2^-t/12.

(a)Find the mass after 12 years.(2)

(b)Find the mass after 24 years.(1)

(c)State what the number 80 represents in this context.(1)

(Total for Question 14 is 4 marks)

Given that y = x^-3/2, find the value of y when x = 4.

(Total for Question 15 is 2 marks)

Simplify fully (a^1/2 x a^5/2) / a². Give your answer in the form aⁿ.

(Total for Question 16 is 3 marks)

Show that 4^-3/2 x 8^2/3 = 1/2

(Total for Question 17 is 3 marks)

A cube-shaped storage box has volume V = 512 cm³. The side length of the cube is given by l = V^1/3.

(a)Work out the side length, l, of the cube.(1)

(b)The total surface area of a cube is given by A = 6l². Work out the total surface area of the box, stating the correct units.(2)

(Total for Question 18 is 3 marks)

Mark scheme · 6.2 Fractional and Negative Indices

Question 1

(a) B1 1 cao
(a) Answer: 1
(b) B1 1/8 oe (0.125)
(b) Answer: 1/8 (0.125)
(c) B1 1/5 oe (0.2)
(c) Answer: 1/5 (0.2)

Question 2

B1 C (1/25) cao
Answer: C) 1/25

Question 3

(a) B1 8 cao
(a) Answer: 8
(b) B1 2 cao
(b) Answer: 2
(c) B1 3 cao
(c) Answer: 3

Question 4

(a) M1 4^1/2 = 2 seen, or 1/(4^1/2) written
(a) A1 1/2 oe (0.5) cao
(a) Answer: 1/2 (0.5)
(b) M1 9^1/2 = 3 and 3³ = 27 (or 9^3/2 = 27) seen
(b) A1 1/27 oe cao
(b) Answer: 1/27

Question 5

(a) B1 x³ cao
(a) Answer: x³
(b) B1 y⁷ cao
(b) Answer: y⁷
(c) B1 m^-6 cao
(c) Answer: m^-6

Question 6

B1 A (x^-3) cao
Answer: A) x^-3

Question 7

(a) M1 8^1/3 = 2 seen
(a) A1 4 cao
(a) Answer: 4
(b) M1 25^1/2 = 5 seen
(b) A1 125 cao
(b) Answer: 125
(c) M1 16^1/4 = 2 seen
(c) A1 1/2 oe (0.5) cao
(c) Answer: 1/2 (0.5)

Question 8

(a) M1 one index correctly found: x⁵ or y^-5
(a) A1 x⁵ y^-5 oe (e.g. x⁵ / y⁵) cao
(a) Answer: x⁵ y^-5 (i.e. x⁵ / y⁵)
(b) M1 2^-2 = 1/4 and/or a^-6 seen
(b) A1 k = 1/4 and n = -6, both correct
(b) Answer: k = 1/4, n = -6

Question 9

M1 1/32 written as 2^-5 or 32 = 2⁵ seen
A1 x = -5 cao
Answer: x = -5

Question 10

(a) B1 x = 49 cao
(a) Answer: x = 49
(b) M1 x^1/3 = 5 (from √25 = 5), or x = 25^3/2 seen
(b) A1 x = 125 cao
(b) Answer: x = 125

Question 11

M1 9^-1/2 = 1/3 oe
M1 3⁴ = 81 seen
A1 cso: (1/3) x 81 = 27 shown, dependent on both method marks
Answer: 27 (shown)

Question 12

M1 27^1/3 = 3 seen
M1 3² = 9, i.e. 27^2/3 = 9 seen
A1 1/9 oe cao
Answer: 1/9

Question 13

(a) B1 200 cao
(a) Answer: 200
(b) M1 200 x 3⁴ with 3⁴ = 81 seen, or correct substitution
(b) A1 16200 cao
(b) Answer: 16200

Question 14

(a) M1 correct substitution t = 12 giving 80 x 2^-1, or 2^-1 = 0.5 seen
(a) A1 40 (g) cao
(a) Answer: 40 grams
(b) B1 20 (g) cao, ft from correct method in (a)
(b) Answer: 20 grams
(c) B1 the initial mass of the sample / the mass when t = 0 oe
(c) Answer: The initial mass of the sample (the mass at t = 0), which is 80 grams.

Question 15

M1 4^1/2 = 2 and/or 4^3/2 = 8 seen
A1 y = 1/8 oe cao
Answer: y = 1/8

Question 16

M1 numerator combined: a^{1/2 + 5/2} = a³
M1 indices subtracted: a^{3 - 2}
A1 a (i.e. a¹) cao
Answer: a (a¹)

Question 17

M1 4^-3/2 = 1/8 oe (from 4^1/2 = 2, 2³ = 8)
M1 8^2/3 = 4 oe (from 8^1/3 = 2, 2² = 4)
A1 cso: (1/8) x 4 = 1/2 shown, dependent on both method marks
Answer: 1/2 (shown)

Question 18

(a) B1 8 (cm) cao
(a) Answer: 8 cm
(b) M1 6 x 8² (= 384) seen, ft from (a)
(b) A1 384 cm², correct units required
(b) Answer: 384 cm²

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Fractional and Negative Indices - Worksheets, Questions and Revision

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6.2 Fractional and Negative Indices

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18