Squares and Square Roots 8 Maths — Revision Notes
Squares are products of a number with itself, and square roots are the inverse operation. Understanding these concepts is essential for algebra and geometr
TL;DR: Squares are products of a number with itself, and square roots are the inverse operation. Understanding these concepts is essential for algebra and ge…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
Squares are products of a number with itself, and square roots are the inverse operation. Understanding these concepts is essential for algebra and geometr
Perfect Squares
- Perfect square is a number that can be expressed as the product of two equal integers
- Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
- First n perfect squares are 1 squared, 2 squared through n squared
- Perfect squares always end in digits 0, 1, 4, 5, 6, or 9
- Perfect squares never end in digits 2, 3, 7, or 8
Properties of Perfect Squares
- Square of even number is always even; of odd number is always odd
- Square of a negative number is positive; (-5) squared equals 25
- Sum of first n odd numbers equals n squared
- Between consecutive perfect squares, no other perfect square exists
- If a number ends in 5, its square ends in 25
Square Roots
- Square root of x is a number that when multiplied by itself gives x
- Symbol is radical sign with value inside
- Positive square root is principal square root
- Both 5 and -5 are square roots of 25, but principal square root is 5
- Square root of perfect square is always an integer
Finding Square Roots
- Prime factorization method: group prime factors in pairs
- Repeated subtraction: subtract odd numbers successively
- Long division method: systematic approach for large numbers
- Estimation: find closest perfect squares to estimate roots
- Using tables: reference square root tables for quick lookup
Square Roots of Rational Numbers and Irrational Numbers
- Square root of a rational number p/q equals square root of p divided by square root of q
- Non-perfect square numbers have irrational square roots
- Square root of 2 is approximately 1.414 and is irrational
- Irrational numbers cannot be expressed as ratio of two integers
- Decimal expansion of irrational square roots is non-terminating and non-repeating
Key Terms
- Perfect Square: Number that equals the product of two equal integers
- Square Root: Number that when multiplied by itself gives the original number
- Irrational Number: Number that cannot be expressed as ratio of two integers
- Principal Square Root: The positive square root of a number
Frequently Asked Questions
What is the square root of 0?
The square root of 0 is 0, since 0 multiplied by 0 equals 0.
Why is the square root of 2 irrational?
The square root of 2 cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating.
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