Volume = 43 πr 3
Cylinder:
Volume = πr 2 h
Cone:
Volume = 13 πr 2 h
15. Statistics & Probability
∑x
Mean: n
Median: Middle value in an ordered list.
Mode: Most frequently occurring value.
Probability: Favorable outcomes
Total outcomes
These notes provide clear definitions, key
Prepare proper notes on squares and square roots according to the CBSE class 8 level.
Chapter: Squares and Square Roots
1. Introduction
A square number is a number that can be written as the square of an integer.
The square root of a number is a value that, when multiplied by itself, gives the original number.
2. Perfect Squares
A number is called a perfect square if it is the square of a whole number.
Examples:
12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36
Properties of Perfect Squares
1. A number ending in 2, 3, 7, or 8 is never a perfect square.
2. A number ending in 0, 1, 4, 5, 6, or 9 can be a perfect square.
3. If a number is a perfect square, its prime factorization will contain even powers of all prime
numbers.
4. The square of an even number is always even, and the square of an odd number is always odd.
5. The number of zeros at the end of a perfect square is always even.
3. Finding the Square of a Number
(i) Using the Identity Formulas
Algebraic identities for squaring numbers:
2 2 2
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, (a + b)2 = a2 + 2ab + b2
(a − b)2 = a2 − 2ab + b2
(a + b)(a − b) = a2 − b2
Example:
Find 422 using the identity.
(40 + 2)2 = 402 + 2(40 × 2) + 22
= 1600 + 160 + 4 = 1764
(ii) Using the Column Method (Multiplication Method)
Multiply the number by itself.
42 × 42 = 1764
4. Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number.
The symbol for the square root is .
Example:
25 = 5, 49 = 7, 100 = 10
Finding Square Roots
(i) By Prime Factorization Method
Steps:
1. Find the prime factorization of the number.
2. Pair the factors.
3. Take one factor from each pair and multiply them.
Example: Find 144.
144 = 2 × 2 × 2 × 2 × 3 × 3
Grouping the factors:
(2 × 2) × (2 × 2) × (3 × 3)
Taking one from each pair:
2 × 2 × 3 = 12
So, 144 = 12.
(ii) By Long Division Method
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,Used for finding square roots of large numbers that are not perfect squares.
Example: Find 529.
1. Pair the digits from the right: 5 29
2. Find the largest number whose square is less than or equal to 5 → 2 × 2 = 4.
3. Subtract 4 from 5 → 1, bring down 29 → 129.
4. Double the quotient (2) → 4.
5. Find the number that makes 4X × X close to 129 → 7 × 7 = 49.
6. Answer is 23.
5. Properties of Square Roots
1. The square root of an even perfect square is always even.
Example: 36 = 6
2. The square root of an odd perfect square is always odd.
Example: 25 = 5
3. A perfect square has a definite square root, but non-perfect squares have an approximate
square root.
Example: 50 ≈ 7.07
6. Estimating Square Roots
·
For non-perfect squares, use the nearest perfect squares for estimation.
Example: Estimate 90.
92 = 81, 102 = 100
16400
Since 90 is closer to 81, we estimate 90 ≈ 9.4.
1504
7. Applications of Square and Square Roots
Geometry: Finding the side of a square when given the area.
Physics: Calculating distances using the Pythagoras theorem.
Engineering & Science: Used in quadratic equations and physics formulas.
8. Important Formulas
1. Squares of First 10 Natural Numbers:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
2. (a+b)^2 = a^2 + 2ab + b^2
3. (a-b)^2 = a^2 - 2ab + b^2
4. (a+b)(a-b) = a^2 - b^2
5. Square root of a product:
a×b= a× b
6. Square root of a fraction:
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, a a
=
b b
This note provides clear explanations, examples, and formulas needed for CBSE Class 8. Let me know if
you need additional details!
Create similar nodes on Exponents and Radicals.
Chapter: Exponents and Radicals
1. Introduction
Exponents represent repeated multiplication of a number.
Example: 23 = 2 × 2 × 2 = 8.
Radicals are the opposite of exponents. A radical represents a root, such as a square root or
cube root 3 .
2. Laws of Exponents
1. Multiplication Law
am × an = am+n
Example: 23 × 24 = 23+4 = 27 = 128
2. Division Law
am
n
= am−n , (where m > n)
a
Example: 56 ÷ 52 = 56−2 = 54 = 625
3. Power of a Power Law
(am )n = am×n
Example: (32 )4 = 32×4 = 38 = 6561
4. Zero Exponent Law
a0 = 1, (where a
= 0)
Example: 70 =1
5. Negative Exponent Law
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Cylinder:
Volume = πr 2 h
Cone:
Volume = 13 πr 2 h
15. Statistics & Probability
∑x
Mean: n
Median: Middle value in an ordered list.
Mode: Most frequently occurring value.
Probability: Favorable outcomes
Total outcomes
These notes provide clear definitions, key
Prepare proper notes on squares and square roots according to the CBSE class 8 level.
Chapter: Squares and Square Roots
1. Introduction
A square number is a number that can be written as the square of an integer.
The square root of a number is a value that, when multiplied by itself, gives the original number.
2. Perfect Squares
A number is called a perfect square if it is the square of a whole number.
Examples:
12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36
Properties of Perfect Squares
1. A number ending in 2, 3, 7, or 8 is never a perfect square.
2. A number ending in 0, 1, 4, 5, 6, or 9 can be a perfect square.
3. If a number is a perfect square, its prime factorization will contain even powers of all prime
numbers.
4. The square of an even number is always even, and the square of an odd number is always odd.
5. The number of zeros at the end of a perfect square is always even.
3. Finding the Square of a Number
(i) Using the Identity Formulas
Algebraic identities for squaring numbers:
2 2 2
Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 6/50
, (a + b)2 = a2 + 2ab + b2
(a − b)2 = a2 − 2ab + b2
(a + b)(a − b) = a2 − b2
Example:
Find 422 using the identity.
(40 + 2)2 = 402 + 2(40 × 2) + 22
= 1600 + 160 + 4 = 1764
(ii) Using the Column Method (Multiplication Method)
Multiply the number by itself.
42 × 42 = 1764
4. Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number.
The symbol for the square root is .
Example:
25 = 5, 49 = 7, 100 = 10
Finding Square Roots
(i) By Prime Factorization Method
Steps:
1. Find the prime factorization of the number.
2. Pair the factors.
3. Take one factor from each pair and multiply them.
Example: Find 144.
144 = 2 × 2 × 2 × 2 × 3 × 3
Grouping the factors:
(2 × 2) × (2 × 2) × (3 × 3)
Taking one from each pair:
2 × 2 × 3 = 12
So, 144 = 12.
(ii) By Long Division Method
Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 7/50
,Used for finding square roots of large numbers that are not perfect squares.
Example: Find 529.
1. Pair the digits from the right: 5 29
2. Find the largest number whose square is less than or equal to 5 → 2 × 2 = 4.
3. Subtract 4 from 5 → 1, bring down 29 → 129.
4. Double the quotient (2) → 4.
5. Find the number that makes 4X × X close to 129 → 7 × 7 = 49.
6. Answer is 23.
5. Properties of Square Roots
1. The square root of an even perfect square is always even.
Example: 36 = 6
2. The square root of an odd perfect square is always odd.
Example: 25 = 5
3. A perfect square has a definite square root, but non-perfect squares have an approximate
square root.
Example: 50 ≈ 7.07
6. Estimating Square Roots
·
For non-perfect squares, use the nearest perfect squares for estimation.
Example: Estimate 90.
92 = 81, 102 = 100
16400
Since 90 is closer to 81, we estimate 90 ≈ 9.4.
1504
7. Applications of Square and Square Roots
Geometry: Finding the side of a square when given the area.
Physics: Calculating distances using the Pythagoras theorem.
Engineering & Science: Used in quadratic equations and physics formulas.
8. Important Formulas
1. Squares of First 10 Natural Numbers:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
2. (a+b)^2 = a^2 + 2ab + b^2
3. (a-b)^2 = a^2 - 2ab + b^2
4. (a+b)(a-b) = a^2 - b^2
5. Square root of a product:
a×b= a× b
6. Square root of a fraction:
Printed using ChatGPT to PDF, powered by PDFCrowd HTML to PDF API. 8/50
, a a
=
b b
This note provides clear explanations, examples, and formulas needed for CBSE Class 8. Let me know if
you need additional details!
Create similar nodes on Exponents and Radicals.
Chapter: Exponents and Radicals
1. Introduction
Exponents represent repeated multiplication of a number.
Example: 23 = 2 × 2 × 2 = 8.
Radicals are the opposite of exponents. A radical represents a root, such as a square root or
cube root 3 .
2. Laws of Exponents
1. Multiplication Law
am × an = am+n
Example: 23 × 24 = 23+4 = 27 = 128
2. Division Law
am
n
= am−n , (where m > n)
a
Example: 56 ÷ 52 = 56−2 = 54 = 625
3. Power of a Power Law
(am )n = am×n
Example: (32 )4 = 32×4 = 38 = 6561
4. Zero Exponent Law
a0 = 1, (where a
= 0)
Example: 70 =1
5. Negative Exponent Law
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