SQUARES AND SQUARE ROOTS

Definitions:

Square: The product of a number multiplied by itself (e.g., 5² = 5 * 5 = 25).
Square Root: The number that, when multiplied by itself, gives the original number (e.g., √25 = 5, as 5 * 5 = 25).

Properties of Squares:

Unit Digit: The unit digit of a perfect square (a square formed by a whole number) depends only on the unit digit of the original number.

Unit digit of original number	Possible unit digits of the square
0	0
1	1
2	4, 6
3	9
4	6
5	5
6	6
7	9
8	4
9	1

Sum of consecutive odd numbers: Any perfect square can be expressed as the sum of consecutive odd numbers, starting from 1.

Examples:

3² = 9 = 1 + 3 + 5
5² = 25 = 1 + 3 + 5 + 7 + 9

Properties of Square Roots:

Irrational Numbers: Most square roots are irrational, meaning they cannot be expressed as a finite decimal or a simple fraction. Examples include √2, √3, and √5.
Perfect Squares: The square root of a perfect square is a rational number.
Approximation Methods: Various methods exist to approximate irrational square roots, such as long division or calculators.

How to Find Square of a Number?

Finding the square of a number involves multiplying the number by itself. Here are the steps to find the square of a number:

Method 1: Using the Multiplication Operation

Choose a Number:
- Select the number for which you want to find the square.
Multiply the Number by Itself:
- Multiply the chosen number by itself.
- Example: To find the square of , calculate .
Write the Result:
- Write down the result.
- Example: The square of is .

Method 2: Using Exponent Notation

Choose a Number:
- Select the number for which you want to find the square.
Use Exponent Notation:
- Express the square using exponent notation. The square of a number $a$ is denoted as $a^{2}$ .
- Example: represents the square of .
Calculate the Result:
- Calculate the result using the exponent notation.
- Example: $5 \times 5 = 25$

Examples:

Square of 7:
- Using multiplication:
- Using exponent notation:
Square of 12:
- Using multiplication:
- Using exponent notation:
Square of 0:
- Using multiplication:
- Using exponent notation:

Square of a Negative Number

The square of a negative number is always positive. This is because squaring involves multiplying a number by itself, and when you square a negative number, the negative sign is squared as well, resulting in a positive value. Here's the explanation:

Example:

Square of $- 3$ :
- Using multiplication:
- Using exponent notation:
Square of $- 5$ :
- Using multiplication:
- Using exponent notation:

Explanation:

When you square a negative number $a$ , you are essentially performing the operation $(- a) \times (- a)$ . The multiplication of two negative numbers results in a positive product. Therefore, the square of a negative number is positive.

General Rule:

For any real number $a$ , $(- a)^{2} = a^{2}$ . The square of the negation of a number is equal to the square of the original number

Square Root of Imperfect Squares

The square root of an imperfect square is an irrational number. An imperfect square is a number whose square root is not a whole number or an integer. When you take the square root of an imperfect square, the result is a non-terminating, non-repeating decimal or an irrational number.

Examples:

Square Root of 2:
- √2 is an example of an imperfect square.
- It cannot be expressed as a fraction of two integers, and its decimal expansion goes on forever without repeating.
Square Root of 5:
- is another example.
- It is an irrational number with a non-repeating and non-terminating decimal expansion.

General Rule:

For any positive integer $n$ that is not a perfect square, $n$ is an irrational number.

Properties of Irrational Numbers:

Non-Repeating Decimal Expansion:
- The decimal expansion of irrational numbers neither repeats nor terminates.
Non-Fractional Form:
- Irrational numbers cannot be expressed as the quotient of two integers.
Examples:
- Other examples of irrational numbers include √3, √7, and $π$ .

Representation:

Imperfect squares have square roots that are not whole numbers, and their representation often involves using the radical symbol √

Properties of Square and Square Roots

Feature	Squares	Square Roots
Definition	Product of a number multiplied by itself (n² = n * n)	Number that, when multiplied by itself, gives the original number (√n = n^(1/2))
Examples	1² = 1, 4² = 16, 9² = 81	√1 = 1, √4 = 2, √9 = 3
Representation	Whole numbers raised to the power of 2	Radical symbol with number inside
Types	Perfect squares (square of a whole number), imperfect squares (not a perfect square)	Rational (for perfect squares), irrational (for most imperfect squares)
Unit Digit	Depends on the unit digit of the original number (specific patterns exist)	No defined pattern for unit digit
Sum of Consecutive Odd Numbers	Can be expressed as the sum of consecutive odd numbers starting from 1 (only for perfect squares)	Not applicable
Divisibility Rules	Divisibility by 2 (for even squares), 4 (for squares ending in 2 or 8), 5 (for squares ending in 5 or 0), 9 (for squares whose digits add up to a multiple of 9)	Not applicable
Operations	Add, subtract, multiply, and divide like any exponents	Add, subtract, multiply, and divide with specific rules (usually involving rationalization for irrational roots)
Applications	Geometry (areas of squares), physics (kinetic energy), finance (calculations involving squares and square roots)	Approximation methods, engineering calculations, scientific and statistical analysis

Square and Square Root Table 1 to 20

Number	Square	Square Root (Approx.)
1	1	1.000
2	4	1.414
3	9	1.732
4	16	2.000
5	25	2.236
6	36	2.449
7	49	2.646
8	64	2.828
9	81	3.000
10	100	3.162
11	121	3.316
12	144	3.464
13	169	3.606
14	196	3.742
15	225	3.873
16	256	4.000
17	289	4.123
18	324	4.243
19	361	4.359
20	400	4.472

Solved Examples of Squares and Square roots

Finding the Square of a Number:

Example: Find the square of 8.
Solution: 8² = 8 * 8 = 64

2. Identifying a Perfect Square:

Example: Is 64 a perfect square?
Solution: Yes, 64 is the square of 8 (8² = 64).

3. Finding the Square Root of a Perfect Square:

Example: Find the square root of 25.
Solution: √25 = 5 (5 * 5 = 25)

4. Approximating the Square Root of an Imperfect Square:

Example: Find the square root of 17 (approximate).
Solution: Using a calculator, √17 ≈ 4.123 (a closer approximation can be found using more advanced methods).

5. Adding and Subtracting Squares:

Example: Add 4² and 9².
Solution: 4² + 9² = 16 + 81 = 97

6. Multiplying and Dividing Squares:

Example: Multiply 3² by 5².
Solution: 3² * 5² = (3 * 3) * (5 * 5) = 9 * 25 = 225
Example: Divide 144 by 4².
Solution: 144 / 4² = 144 / (4 * 4) = 144 / 16 = 9

7. Applying Squares and Square Roots in Real-World Problems:

Example: Calculate the area of a square with a side length of 6 cm.
Solution: Area = side² = 6² = 6 * 6 = 36 cm²

8. Using Properties of Squares and Square Roots:

Example: Identify the divisibility rule for squares ending in 5.
Solution: Squares ending in 5 always have a unit digit of 5

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SQUARES AND SQUARE ROOTS

SQUARES AND SQUARE ROOTS

Method 1: Using the Multiplication Operation

Method 2: Using Exponent Notation

Examples:

Example:

Explanation:

General Rule:

Examples:

General Rule:

Properties of Irrational Numbers:

Representation:

Square and Square Root Table 1 to 20

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