Perfect Squares

Procedure to check whether a given natural number is a perfect squares or not.

Step I : Obtain the natural number.

Step II : Write the number as a product of prime factors.

Step III : Group the factors in pairs in such a way that both the factors in each pair are equal.

Step IV: See whether some factor is left over or not. If no factor is left over in the grouping, then the given number is a perfect square. Otherwise, it is not a perfect-square.

Step V: To obtain the number whose square is the given number taken over one factor from each group and multiply them.

Examples on perfect-square

1) Is 225 a perfect-square? If so, find the number whose square is 225.

Solution :
Resolving 225 into prime factors, we obtain
225 = 3 x 3 x 5 x 5
Grouping the factors in pairs in such a way that both the factors in each pair are equal, we have
225 = ( 3 x 3 ) x ( 5 x 5 )
Clearly, 225 can be grouped into pairs of equal factors and no factor is left over.
Hence, 225 is a perfect-square.
Again, 225 = (3 x 5) x (3 x 5)
= 15 x 15 = 15
2
So, 225 is the square of 15.
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2) Is 150 a perfect-square? If so,find the number whose square is 150.

Solution :
Resolving 150 into prime factors, we obtain
150 = 2 x 3 x 5 x 5
Grouping the factors in pairs in such a way that both the factors in each pair are equal, we have
225 = 2 x 3 x ( 5 x 5 )
Clearly, 150 can be grouped into pairs of equal factors.2 and 3 factors are left over.
Hence, 150 is not a perfect-square.
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Practice Problems

Q.1 Check whether the following numbers are perfect- squares or not, give reason.

1) 250 2) 289 3) 1024 4) 1156 5) 1000

Q.2 The following numbers are perfect-squares,find whose perfect-squares are those.

1) 2 x 3 x 3 x 2 x 5 x 5
2) 7 x 7 x 2 x 11 x 2 x 11
3) 2 x 3 x 3 x 2 x 7 x 7

Squares and Square roots

Introduction of Squares and Square Roots
Perfect Squares or not
Properties of Square Numbers
Short cut method to find squares
Introduction of Square Roots
Properties of Square Roots
Square root by Prime factorization method
Square root by long division method
Square root of rational numbers
Square root of Decimals
Square root by estimation method

From squares and square roots to Exponents

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