This checkpoint assesses your understanding of index notation, prime factors, the index laws, zero and negative indices. It covers Lessons 6–10.
❓ Multiple Choice (10 questions)
1. Evaluate $2^4$.
2. What is $10^0$?
3. Simplify $3^5 \times 3^2$.
4. Simplify $x^8 \div x^3$.
5. Simplify $(2^3)^2$.
6. Evaluate $2^{-3}$.
7. What is the prime factorisation of $72$?
8. Simplify $(3a^2)^3$.
9. Simplify $2^5 \times 2^{-3} \div 2^2$.
10. Evaluate $(2 \times 10^3) \times (4 \times 10^2)$.
✍ Short Answer (4 questions)
11. Evaluate and simplify.
(a) $5^3$ (1 mark)
(b) $4^7 \div 4^4$ (1 mark)
(c) $3^{-2}$ (1 mark)3 MARKS
12. Simplify fully.
(a) $(2^3)^2$ (1 mark)
(b) $x^5 \times x^3 \div x^2$ (1 mark)
(c) $(3a^2b) \times (2ab^3)$ (2 marks)4 MARKS
13. Prime factors and surds.
(a) Find the prime factorisation of $180$. (2 marks)
(b) Use your answer to find $\sqrt{180}$ in simplified surd form. (2 marks)4 MARKS
14. Mixed problems.
(a) Simplify $(2^4 \times 2^{-2})^3 \div 2^4$. (2 marks)
(b) Evaluate $(3 \times 10^2)^2 \div (2 \times 10^3)$. (2 marks)
(c) If $2^a \times 2^3 = 2^5$, find $a$. (1 mark)5 MARKS
1. B — $2^4 = 2 \times 2 \times 2 \times 2 = 16.
2. B — Any non-zero number to the power of 0 equals 1.
3. A — $3^5 \times 3^2 = 3^{5+2} = 3^7.
4. A — $x^8 \div x^3 = x^{8-3} = x^5.
5. B — $(2^3)^2 = 2^{3 \times 2} = 2^6 (which equals 64).
6. C — $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}.
7. B — $72 = 8 \times 9 = 2^3 \times 3^2 = 2^3 \times 3^2.
8. C — $(3a^2)^3 = 3^3 \times (a^2)^3 = 27 \times a^6 = 27a^6.
9. B — $2^5 \times 2^{-3} \div 2^2 = 2^{5-3-2} = 2^0 = 1.
10. A — $(2 \times 10^3) \times (4 \times 10^2) = 8 \times 10^{3+2} = 8 \times 10^5.
Q11 (3 marks): (a) $5^3 = 5 \times 5 \times 5 = 125$ [1]. (b) $4^7 \div 4^4 = 4^{7-4} = 4^3 = 64$ [1]. (c) $3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$ [1].
Q12 (4 marks): (a) $(2^3)^2 = 2^{3 \times 2} = 2^6 = 64$ [1]. (b) $x^5 \times x^3 \div x^2 = x^{5+3-2} = x^6$ [1]. (c) $3a^2b \times 2ab^3 = 6 \times a^{2+1} \times b^{1+3} = 6a^3b^4$ [2].
Q13 (4 marks): (a) $180 = 2 \times 90 = 2 \times 2 \times 45 = 2^2 \times 3^2 \times 5 = 2^2 \times 3^2 \times 5$ [2]. (b) $\sqrt{180} = \sqrt{2^2 \times 3^2 \times 5} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5}$ [2].
Q14 (5 marks): (a) $(2^4 \times 2^{-2})^3 \div 2^4 = (2^2)^3 \div 2^4 = 2^6 \div 2^4 = 2^2 = 4$ [2]. (b) $(3 \times 10^2)^2 \div (2 \times 10^3) = 9 \times 10^4 \div (2 \times 10^3) = 4.5 \times 10^1 = 45$ [2]. (c) $2^a \times 2^3 = 2^5 \Rightarrow 2^{a+3} = 2^5 \Rightarrow a + 3 = 5 \Rightarrow a = 2$ [1].
Tick when you have finished all questions and checked your answers.