Squares, Cubes & Roots

Techniques to handle powers and roots in numericals.

1. Squaring Numbers

A. Numbers ending in 5

Rule: For a number $(n)5$: First part is $n \times (n+1)$, last part is always $25$.

Square of 35:
First part: 3 x 4 = 12
Last part: 25
Answer: 1225

B. Near 50 Base Method

Rule: Compare with 25.

Square of 48 (2 less than 50):
First part: 25 - 2 = 23
Last part: $2^2$ = 04 (Must be 2 digits)
Answer: 2304

2. Square Roots (Approximation)

Non-Perfect Squares Formula

This is crucial for Physics/Chem problems (e.g., RMS speed, Equilibrium constant).

$$ \sqrt{x \pm y} \approx \sqrt{x} \pm \frac{y}{2\sqrt{x}} $$

Where $x$ is the nearest perfect square and $y$ is the difference.

Find $\sqrt{27}$:
Nearest square is 25 ($x=25, y=2$)
$$ \sqrt{25 + 2} \approx 5 + \frac{2}{2 \times 5} $$ $$ \approx 5 + \frac{2}{10} = 5.2 $$ (Actual: 5.196)

Find $\sqrt{34}$:
Nearest square is 36 ($x=36, y=2$)
$$ \sqrt{36 - 2} \approx 6 - \frac{2}{2 \times 6} $$ $$ \approx 6 - \frac{1}{6} \approx 6 - 0.16 = 5.84 $$ (Actual: 5.83)

3. Cube Roots (Estimation)

Last Digit Trick (Perfect Cubes)

Look at the last digit of the number to determine the last digit of the root.

Ends in 1 $\rightarrow$ 1
Ends in 4 $\rightarrow$ 4
Ends in 5 $\rightarrow$ 5
Ends in 6 $\rightarrow$ 6
Ends in 9 $\rightarrow$ 9
Ends in 2 $\leftrightarrow$ 8 (Flip)
Ends in 3 $\leftrightarrow$ 7 (Flip)

Find $\sqrt[3]{12167}$:
1. Last digit is 7 $\rightarrow$ Root ends in 3.
2. Ignore last 3 digits (167).
3. Remaining is 12. Closest cube below 12 is $2^3=8$.
Answer: 23

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