Topic Review — Indices & Square Numbers

Review Questions

1. Index Notation — Write and Evaluate Fluency

   1. Write 5 × 5 × 5 × 5 in index notation.
   2. Write 2 × 2 × 2 × 2 × 2 in index notation.
   3. Evaluate 3⁴.
   4. Evaluate 2⁷.
   5. Evaluate 10⁴.
   6. Write 6³ as a repeated multiplication, then evaluate.
   7. What is the value of 5⁰?
   8. What is the value of 9¹?

2. Square Numbers and Square Roots Fluency

   1. Calculate 11².
   2. Calculate 13².
   3. Find √64.
   4. Find √196.
   5. Is 150 a perfect square? Explain.
   6. Between which two consecutive whole numbers does √60 lie?
   7. Between which two consecutive whole numbers does √150 lie?

3. Prime Factorisation Fluency

   1. Write the prime factorisation of 24.
   2. Write the prime factorisation of 60.
   3. Write the prime factorisation of 36 in index form.
   4. Write the prime factorisation of 100 in index form.
   5. Is 41 prime or composite?
   6. List all prime numbers between 20 and 40.

4. HCF, LCM and Number Laws Understanding

   1. Find HCF(18, 24).
   2. Find LCM(6, 9).
   3. True or False: 5² = 2⁵. Explain.
   4. Which law is used in: 7 × (10 + 4) = 7 × 10 + 7 × 4?
   5. Which law is used in: (3 + 9) + 1 = 3 + (9 + 1)?
   6. True or False: 15 ÷ 3 = 3 ÷ 15. Give a reason.
   7. Use the distributive law to calculate 8 × 99 mentally.

5. Comparing and Reasoning with Powers Understanding

   1. Which is larger: 4³ or 3⁴? Show working.
   2. What is √(10²)? What law does this illustrate?
   3. A number has prime factorisation 2³ × 5². What is the number?
   4. Explain why 2¹⁰ ≠ 20.
   5. The area of a square is 169 cm². What is the side length? What is the perimeter?

6. Problem Solving — Mixed Problem Solving

   1. A colony of bacteria triples every hour. Starting with 1 bacterium, write an expression for the number after 5 hours. Calculate the result.
   2. Two traffic lights change every 18 seconds and 24 seconds respectively. Both change at the same time at 8:00 am. After how many seconds will they next change together? (Use LCM.)
   3. Use the commutative and associative laws to find the easiest way to calculate: 4 × 17 × 25. Show your working.
   4. A square paddock has an area of 225 m². Fencing costs $12 per metre. What is the total cost to fence the paddock?
   5. Explain the difference between 3⁴ and 4³. Are they equal? Which is larger?

7. Order of operations with indices (BODMAS): Fluency

   1. 3 + 2³
   2. 5² − 4 × 3
   3. (2 + 3)²
   4. √49 × 3
   5. 2³ + 3² − 1

8. HCF and LCM word problems: Understanding

   1. Hotdog rolls come in packs of 6. Hotdogs come in packs of 8. What is the minimum number of each pack you need to buy so none are left over?
   2. Find the HCF of 36 and 48. Use it to simplify the fraction 3648.
   3. Find the LCM of 4, 6, and 9.

9. Index laws exploration: Understanding

   1. Calculate 2³ × 2². How does the answer relate to 2⁵?
   2. Calculate 3⁴ ÷ 3². How does the answer relate to 3²?
   3. Without calculating, predict: 5⁶ ÷ 5⁶. Explain using the zero index law.

10. A checkerboard has 64 squares. Suppose you place 1 grain of rice on the 1st square, 2 on the 2nd, 4 on the 3rd, doubling each time. Problem Solving

    1. Write the number of grains on the 5th square using index notation.
    2. How many grains are on the 8th square?
    3. On which square are there exactly 64 grains? Express 64 as a power of 2.