Indices

14 Grade 10 Essential Mathematics questions on Indices, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   1 mark

   The value of $2^{-3}$ is:

   A. −6

   B. $\frac{1}{8}$

   C. $\frac{1}{6}$

   D. −8

   (1 mark)

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2. Question 2

   1 mark

   The thickness of a polythene sheet used in a greenhouse in Naivasha is 0.000042 m. Written in standard form, this measurement is:

   A. $4.2\times 10^{-5}$ m

   B. $42\times 10^{-6}$ m

   C. $4.2\times 10^{-4}$ m

   D. $4.2\times 10^5$ m

   (1 mark)

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3. Question 3

   3 marks

   Simplify $3^4\times 3^{-2}÷3^3$, giving your answer as a fraction in its simplest form.

   (3 marks)

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4. Question 4

   3 marks

   The mass of one grain of sorghum is $2.8\times 10^{-5}$ kg and a store holds $2\times 10^6$ such grains.

   (a) Calculate the total mass of the grains in the store, giving your answer in standard form.

   (b) Write the total mass as an ordinary number in kilogrammes.

   (3 marks)

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5. Question 5

   1 mark

   Simplify $3x^4\times 4x^2$.

   A. 12$x^2$

   B. 7$x^6$

   C. 12$x^8$

   D. 12$x^6$

   (1 mark)

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6. Question 6

   3 marks

   The Kenyan portion of Lake Victoria covers about $4.1\times 10^3$ km² while Lake Naivasha covers about $1.4\times 10^2$ km². Calculate how many times as large the Kenyan portion of Lake Victoria is as Lake Naivasha, giving your answer to the nearest whole number.

   (3 marks)

   Marking scheme

   Write as a division: $(4.1\times 10^3)÷(1.4\times 10^2)$ (1 mark). Divide the numbers and subtract the indices: $(4.1÷1.4)\times 10^{3-2}=2.93\times 10$ (1 mark). = 29.3, so it is about 29 times as large (1 mark).

7. Question 7

   4 marks

   Simplify $(2a^{2}b{)}^3\times 3a{b}^4÷4a^3{b}^2$.

   (4 marks)

   Marking scheme

   $(2a^{2}b{)}^3=8a^6{b}^3$ (1 mark). $8a^6{b}^3\times 3a{b}^4=24a^7{b}^7$ (1 mark). Divide the coefficients: $24÷4=6$ (1 mark). Subtract the indices: a^(7−3) b^(7−2), giving 6$a^4$$b^5$ (1 mark).

8. Question 8

   5 marks

   A fish farmer in Kirinyaga stocks a pond with 5 000 tilapia fingerlings. Under ideal conditions the number doubles every week.

   (a) Write an expression in index form for the number of fingerlings after n weeks.

   (b) Calculate the number after 6 weeks.

   (c) State one assumption made in using this expression.

   (5 marks)

   Marking scheme

   (a) Number after n weeks = $5000\times 2^n$ (1 mark). (b) Substitute n = 6: $5000\times 2^6$ (1 mark); $2^6=64$ (1 mark); $5000\times 64=320000$ fingerlings (1 mark). (c) Any one assumption (1 mark): none of the fingerlings die; feeding and oxygen remain sufficient; the pond does not become overcrowded; the doubling rate stays exactly the same each week.

9. Question 9

   1 mark

   Simplify $(2x^3{)}^4$.

   A. 2$x^{12}$

   B. 8$x^{12}$

   C. 16$x^7$

   D. 16$x^{12}$

   (1 mark)

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10. Question 10

    1 mark

    Simplify $(a^7{b}^3)÷(a^2{b}^3)$.

    A. $a^9$

    B. $a^5$

    C. $a^5$$b^6$

    D. $a^5$$b^3$

    (1 mark)

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11. Question 11

    1 mark

    Given that 3ˣ = 81, the value of x is:

    A. 4

    B. 9

    C. 27

    D. 3

    (1 mark)

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12. Question 12

    1 mark

    Evaluate 64^(2/3).

    A. 512

    B. 16

    C. 8

    D. 128

    (1 mark)

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13. Question 13

    4 marks

    Using $5^3÷5^3$, explain why any non-zero number raised to the power zero is equal to 1.

    (4 marks)

    Marking scheme

    In expanded form $5^3÷5^3=(5\times 5\times 5)/(5\times 5\times 5)$ (1 mark). The numerator and denominator are equal, so the value is 1 (1 mark). By the division law, $5^3÷5^3=5^{3-3}=5^0$ (1 mark). The two results must agree, so $5^0=1$, and in general $a^0=1$ for any a ≠ 0 (1 mark; the condition a ≠ 0 need not be stated for the mark).

14. Question 14

    2 marks

    Simplify $(p^6{q}^2)÷(p^2{q}^5)$, giving your answer with positive indices only.

    (2 marks)

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