Day 17: Squares & Square Roots | Stage Middle (6-8) Mathematics | Apex Institute of Maths and Sciences

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Day 17: Squares & Square Roots | Stage Middle (6-8) Mathematics | Apex Institute of Maths and Sciences

1 The Quest: Unlocking Roots

Imagine a square floor made of tiles. If you know the total number of tiles, finding the number of tiles on one side is like finding the Square Root! It is the “reverse” of squaring a number.

Example: $4^2 = 16$, so the square root of $16$ (written as $\sqrt{16}$) is $4$.
Introduction to Cube Roots: Similarly, if $2 \times 2 \times 2 = 8$, then the cube root of $8$ (written as $\sqrt[3]{8}$) is $2$.
2 Power-Ups: Mastery Tools
1. Prime Factorization: Break the number into prime factors and pick one from each pair.
2. Long Division: Best for large numbers where factors are hard to find!
Long Division Step: Group digits in pairs from right to left (e.g., $5,29 \rightarrow \overline{5}\overline{29}$). Find the largest square less than the first group and proceed!
3 Mini-Boss Battles: Real Life
Scenario 1: The Garden Designer
A gardener has 625 saplings and wants to plant them in a square grid (equal rows and columns). By calculating $\sqrt{625} = 25$, he knows he needs 25 rows!
Scenario 2: The Gift Box
A cubic box has a volume of $216\text{ cm}^3$. To find the length of one side, we find the cube root: $\sqrt[3]{216} = 6\text{ cm}$.
4 Home Quests: Family Fun
Task 1: The Kitchen Count
Find a square object at home (like a tile or a coaster). Measure one side. Multiply it by itself to find the “Area” and explain to your parents how the side is the square root of that area.
Task 2: Pattern Hunter
Write down squares of numbers from 1 to 15 on a paper. Decorate the perfect squares with a star and show them to your siblings!
5 Final Boss: Practice Test
EASY

1. What is the square root of 144?

Magic Solution: Since $12 \times 12 = 144$, the square root is 12.
EASY

2. Which of these is a perfect square?

Magic Solution: 81 is a perfect square because $9 \times 9 = 81$.
EASY

3. Find the value of $\sqrt[3]{27}$.

Magic Solution: $3 \times 3 \times 3 = 27$, so the cube root is 3.
EASY

4. The square root of 1 is:

Magic Solution: $1 \times 1 = 1$, so $\sqrt{1} = 1$.
MODERATE

5. Find the square root of 625 using prime factorization.

Magic Solution: $625 = 5 \times 5 \times 5 \times 5$. Grouping in pairs: $(5 \times 5) \times (5 \times 5)$. Root = $5 \times 5 = 25$.
MODERATE

6. What is the smallest number by which 32 must be multiplied to make it a perfect square?

Magic Solution: $32 = 2 \times 2 \times 2 \times 2 \times 2$. One ‘2’ is left without a pair. So, multiply by 2.
MODERATE

7. If $x^2 = 0.04$, what is $x$?

Magic Solution: $0.2 \times 0.2 = 0.04$.
MODERATE

8. Find $\sqrt{1764}$ using long division.

Magic Solution: Dividing 1764 by long division method gives 42 ($40^2=1600$, $50^2=2500$, ends in 4, so $42 \times 42 = 1764$).
COMPLEX

9. A square area is $1296\text{ m}^2$. Find the cost of fencing it at ₹10 per meter.

Magic Solution: Side = $\sqrt{1296} = 36\text{ m}$. Perimeter = $4 \times 36 = 144\text{ m}$. Cost = $144 \times 10 = ₹1440$.
COMPLEX

10. Find the smallest square number divisible by 4, 9, and 10.

Magic Solution: LCM of 4, 9, 10 is 180. $180 = 2 \times 2 \times 3 \times 3 \times 5$. To make it a square, multiply by 5: $180 \times 5 = 900$.

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