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HIGHER

4.2H Indices: Higher Tier Practice

EDEXCEL Edexcel 1MA1 · Calculator allowed · about 60 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Work out 3² * 3^-1.

(Total for Question 1 is 1 mark)

Simplify x³ * x^-5.

(Total for Question 2 is 2 marks)

Write 0.00072 in standard form (a x 10ⁿ where 1 ≤ a < 10).

(Total for Question 3 is 2 marks)

Solve 2^x = 32.

(Total for Question 4 is 2 marks)

Simplify (x² y^-3)² / (x^-1 y). Give your answer with positive or negative powers as appropriate.

(Total for Question 5 is 3 marks)

Show that 2ⁿ * 5ⁿ = 10ⁿ for any integer n.

(Total for Question 6 is 1 mark)

Evaluate 27^2/3.

(Total for Question 7 is 2 marks)

Solve 4^x-1 = 8. Give your answer as a fraction or mixed number if needed.

(Total for Question 8 is 3 marks)

Show that (x³)^1/2 * (x²)^1/3 = x^13/6, stating any restriction on x if necessary.

(Total for Question 9 is 3 marks)

Simplify (16 x⁴)^3/2. Give your answer with integer coefficients and powers.

(Total for Question 10 is 3 marks)

Solve (1/4) * 2^3x = 8. Give your answer as a fraction.

(Total for Question 11 is 3 marks)

Simplify (2 x^-1/3)³ * (4 x^2/3)². Give your answer simplified as a single term.

(Total for Question 12 is 3 marks)

Show that x^2/3 / x^1/6 = x^1/2. State any restriction on x.

(Total for Question 13 is 3 marks)

A culture contains 500 bacteria. The population doubles every 3 hours. Write an expression for the number of bacteria after t hours, then work out how many after 18 hours.

(Total for Question 14 is 3 marks)

Mark scheme · 4.2H Indices: Higher Tier Practice

Question 1

B1 3 cao
Answer:3
(3² * 3^-1 = 3^2-1 = 3¹ = 3 )

Question 2

M1 subtract exponents: 3 - 5 = -2
A1 x^-2 cao
Answer: x^-2

Question 3

M1 move decimal point 4 places to the right to get 7.2
A1 7.2 x 10^-4 cao
Answer: 7.2 x 10^-4

Question 4

M1 recognise 32 = 2⁵ or take logs
A1 x = 5 cao
Answer: 5

Question 5

M1 apply power of a product: (x² y^-3)² = x⁴ y^-6
M1 divide by x^-1 y gives x^4-(-1) y^-6-1 or equivalent
A1 x⁵ y^-7 cao
Answer: x⁵ y^-7

Question 6

C1 2ⁿ * 5ⁿ = (2*5)ⁿ = 10ⁿ cso
Answer:10ⁿ
(Because 2ⁿ * 5ⁿ = (2*5)ⁿ = 10ⁿ.)

Question 7

M1 interpret 27^2/3 = (cube root of 27)² or (27²)^1/3
A1 9 cao
Answer:9
(since cube root of 27 is 3, 3² = 9)

Question 8

M1 write 4 and 8 as powers of 2: 4^x-1 = (2²)^x-1 = 2^2x-2, 8 = 2³
M1 equate exponents: 2x - 2 = 3
A1 x = 5/2 cao
Answer: 5/2

Question 9

M1 apply power rule: (x³)^1/2 = x^3/2 and (x²)^1/3 = x^2/3
M1 add exponents: 3/2 + 2/3 = (9+4)/6 = 13/6
A1 x^13/6 cao; state x ≥ 0 or work in real principal values
Answer:x^13/6
(Valid for x ≥ 0 if working with real principal fractional powers.)

Question 10

M1 use (ab)^k = a^k b^k and 16^3/2 = (16^1/2)³ = 4³
M1 x^{4 * 3/2} = x⁶
A1 64 x⁶ cao
Answer: 64 x⁶

Question 11

M1 write 1/4 as 2^-2: 2^3x * 2^-2 = 2³
M1 combine exponents: 2^3x-2 = 2³ so 3x - 2 = 3
A1 x = 5/3 cao
Answer: 5/3

Question 12

M1 expand powers: (2³)(x^-1/3 * 3) = 8 x^-1 and (4²)(x^{2/3 *2}) = 16 x^4/3
M1 multiply coefficients 8*16 = 128 and add exponents -1 + 4/3 = 1/3
A1 128 x^1/3 cao
Answer: 128 x^1/3

Question 13

M1 subtract exponents: 2/3 - 1/6 = 4/6 - 1/6 = 3/6
M1 3/6 simplifies to 1/2
A1 x^1/2 cao; state x > 0 for real principal roots (division by x^1/6 requires x ≠ 0)
Answer:x^1/2
(Valid for x > 0 when interpreting fractional powers as principal real roots, since x^1/6 in the denominator cannot be zero.)

Question 14

M1 write expression 500 * 2^t/3 or 500 * 2^{floor?} Accept 500 * 2^t/3 as continuous model
M1 substitute t = 18: exponent 18/3 = 6, so 500 * 2⁶
A1 32000 cao
Answer:Number after t hours: 500 * 2^t/3.
Number after 18 hours: 500 * 2⁶ = 500 * 64 = 32000.

Indices: Higher Tier Practice - Worksheets, Questions and Revision