Practice Maths

L39 — Square Numbers & Square Roots

Key Terms

square number
The result of multiplying a whole number by itself. Written as n2 and read “n squared”. E.g. 72 = 49.
perfect square
A whole number that is a square number. The first fifteen are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
square root
The inverse of squaring. √m is the non-negative number n such that n2 = m. E.g. √49 = 7 because 7 × 7 = 49.
cube number
The result of multiplying a whole number by itself twice more: n3 = n × n × n. E.g. 43 = 64.

Square Numbers and Square Roots

A square number is the result of multiplying a whole number by itself: n2 = n × n.

The square root (√) is the inverse operation. If n2 = m, then √m = n.

At Year 7, √ always gives a non-negative result.

Hot Tip: Not every number is a perfect square. The perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225… If a number is not in this list, its square root is not a whole number.

Worked Examples

Example 1: Find √144.

Ask: what number times itself equals 144? 12 × 12 = 144. Answer: √144 = 12

Example 2: Is 72 a perfect square?

82 = 64, 92 = 81. Since 72 is between 64 and 81 but not equal to either, 72 is not a perfect square. √72 is between 8 and 9.

Square Numbers: Making Squares With Dots

Why are they called “square” numbers? Because you can arrange that many dots into a perfect square shape. 4 dots makes a 2 × 2 square. 9 dots makes a 3 × 3 square. 25 dots makes a 5 × 5 square. The pattern is n2 = n × n, and you can picture it geometrically every time.

This geometric connection is why area problems involve squared units. A square room that is 5 m on each side has an area of 52 = 25 m2. “Square metres” literally means “metres squared.”

The Square Root: Undoing the Square

Square root is the inverse of squaring — just like subtraction undoes addition, and division undoes multiplication. If 42 = 16, then √16 = 4. You’re asking: “What number, multiplied by itself, gives this result?”

Remember: The 15 perfect squares up to 225 are worth memorising: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. If you know these, you can instantly answer any √ question involving perfect squares — no calculator needed.

Non-Perfect Squares: Between Two Values

Most numbers are not perfect squares. For example, √50 is not a whole number. But you can estimate it: 72 = 49 and 82 = 64. Since 50 is between 49 and 64, √50 is between 7 and 8 — closer to 7. With a calculator: √50 ≈ 7.07.

Estimating square roots between two whole numbers is a useful skill for checking calculator answers and for applying Pythagoras’ theorem.

Real-World Connections

Square roots appear naturally in many areas:

  • Construction: if you know a square room has area 36 m2, its side length is √36 = 6 m.
  • Pythagoras: finding diagonal lengths requires square roots.
  • Physics: speed from kinetic energy involves a square root.
Common Mistake: Confusing √9 with 9 ÷ 2. The square root of 9 is 3 (because 3 × 3 = 9), not 4.5. The square root symbol is not the same as dividing by 2. Always ask: “What times itself gives this number?”

  1. Calculate Square Numbers

    Calculate each square number.

    1. 32
    2. 72
    3. 12
    4. 52
    5. 82
    6. 22
    7. 62
    8. 42

  2. Find Square Roots of Perfect Squares

    Find the square root of each perfect square.

    1. √36
    2. √4
    3. √64
    4. √25
    5. √9
    6. √49
    7. √1
    8. √16

  3. Identify Perfect Squares

    1. From the list: 5, 9, 14, 16, 20, 25, 30, 36, 40, 49 — which are perfect squares?
    2. Is 50 a perfect square? Explain.
    3. Is 100 a perfect square? Explain.
    4. Is 200 a perfect square? Explain.
    5. List all perfect squares between 50 and 150.

  4. Estimate Square Roots

    For each number, state which two consecutive whole numbers its square root lies between.

    1. √50
    2. √20
    3. √80
    4. √130
    5. √200
    6. √75
    7. √45
    8. √110

  5. Squaring and Square Rooting as Inverse Operations

    1. What is √(62)? What do you notice?
    2. What is (√49)2? What do you notice?
    3. True or False: √(n2) = n for all positive integers n.
    4. True or False: (√n)2 = n for all perfect squares n.
    5. Explain in your own words why squaring and taking the square root are inverse operations.

  6. Problem Solving with Square Numbers and Roots

    1. A square garden has an area of 49 m². What is the side length?
    2. A square tile has an area of 144 cm². What is the side length?
    3. Square tiles each have an area of 25 cm². How many tiles are needed to cover a floor of area 900 cm²?
    4. A right triangle has legs of 3 cm and 4 cm. The hypotenuse satisfies c2 = 32 + 42. Calculate c2, then find c.
    5. A square field has perimeter 60 m. What is its area?

  7. Cube Numbers

    A cube number is n × n × n = n3. Calculate the first 8 cube numbers.

    1. 13
    2. 23
    3. 33
    4. 43
    5. 53
    6. 63
    7. 73
    8. 83

  8. Squares and Roots — Mixed Table

    Complete the table. Each row has one expression — calculate its value.

      Expression Value
    (a)92 
    (b)√100 
    (c)112 
    (d)√144 
    (e)102 
    (f)√121 
    (g)122 
    (h)√169 

  9. Pythagoras Connection

    For each right triangle, the sides satisfy a2 + b2 = c2. Find the missing side length.

    6 8 c = ? (a)
    5 12 c = ? (b)
    8 15 c = ? (c)
    a = ? 6 10 (d)
    5 b = ? 13 (e)
    1. a = 6, b = 8. Find c.
    2. a = 5, b = 12. Find c.
    3. a = 8, b = 15. Find c.
    4. c = 10, b = 6. Find a.
    5. c = 13, a = 5. Find b.

  10. Real-World Problem Solving with Squares and Roots

    1. A school hall has a square floor area of 196 m². What is the side length of the hall?
    2. A farmer wants to fence a square paddock. The paddock has an area of 2500 m². How much fencing is needed?
    3. A square tablecloth has a perimeter of 280 cm. What is its area?
    4. A computer monitor has a square display with area 2025 cm². What is the diagonal length? (Hint: the diagonal of a square with side s is s√2 ≈ 1.41 × s. Round to 1 decimal place.)
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