Using the prime factorization method, find which of the following numbers are perfect squares 5625

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Question 3 Squares and Square Roots Exercise 3A

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Answer:

Given 5625,

A perfect square can always be expressed as a product of pairs of equal factors.

Now resolve 5625 into prime factors, we get

5625 = 225 X 25 = 9 X 25 X 25 = 3 X 3 X 5 X 5 X 5 X 5 = 3 X 5 X 5 X 3 X 5 X 5

= 75 X 75 = (75)2

Hence, 75 is the number whose square is 5625

∴ 5625 is a perfect square.

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Using the prime factorisation method, find which of the following numbers are perfect squares: i 441 ii 576 iii 11025 iv 1176 v 5625 vi 9075 vii 4225 viii 1089

Solution

(i) 441 = $3\times 3\times 7\times 7$

$=(3{)}^2\times (7{)}^2=(21{)}^2$
$\therefore$ 441 is a perfect square.
(ii) 576= $2\times 2\times 2\times 2\times 2\times 2\times 3\times$

=$(2{)}^2\times (2{)}^2\times (2{)}^2\times (3{)}^2$
=$(2\times 2\times 2\times 3{)}^2=(24{)}^2$
$\therefore$ 576 is a perfect square.
(iii) 11025= $3\times 3\times 5\times 5\times 7\times 7$

=$(3{)}^2\times (5{)}^2\times (7{)}^2$
=$3\times 5\times 7{)}^2=(105{)}^2$
(iv) 1176 = $2\times 2\times 2\times 3\times 7\times 7$

=$(2{)}^2\times 2\times 3\times (7{)}^2$ 1176 is not a perfect square as it cannot be expressed as the product of pair of equal factors

(v) 5625 = $3\times 3\times 5\times 5\times 5\times$

=$(3{)}^2\times (5{)}^2\times (5{)}^2$
=$3\times 5\times 5{)}^2=(75{)}^2$
$\therefore$ 5625 is a perfect square.
(vi) 9075 = $3\times 5\times 5\times 11\times 11$

= $3\times (5{)}^2\times (11{)}^2$
$\therefore$ 9075 is not a perfect square as it cannot be expressed as a product of pair of equal factors
(vii) 4225= $5\times 5\times 13\times 13$

=$(5{)}^2\times (13{)}^2$
=$(5\times 13{)}^2=(65{)}^2$
$\therefore$ 4225 is a perfect square.
(viii) 1089= $3\times 3\times 11\times 11$

=$(3{)}^2\times (11{)}^2$
=$(3\times 11{)}^2=(33{)}^2$
$\therefore$ 1089 is a perfect square.

Mathematics

Secondary School Mathematics VIII

Standard VIII