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

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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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