343 = (7*7)*7
Since 7 does not form a pair we multiply 343 with 7 i.e.
343*7 = 2401 = (7*7)(7*7) = 492
Thus square root of 2401 is 49.
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University of Houston
Samuel D.
Algebra
7 months, 3 weeks ago
We don’t have your requested question, but here is a suggested video that might help.
Find the smallest number that should be multiplied by 5408 to make it a perfect square.
The following steps will be useful to find the least number which has to multiplied by the given number to get a perfect square.
1. Decompose the given numbers into its prime factors.
2. Write the prime factors as pairs such that each pair has two same prime factors.
3. Find the prime factor which does not occur in pair. That is the least number to be multiplied by the given number to get a perfect square.
Example 1 :
Find the least number multiplied by 200 to get a perfect square.
Solution :
Decompose 200 into its prime factors.
Prime factors of 200 :
200 = 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5
= (2 ⋅ 2) ⋅ 2 ⋅ (5 ⋅ 5)
The prime factor 2 does not occur in pair.
So, '2' is the least number to be multiplied by 200 to get a perfect square.
Justification :
√[2(200)] = √[2(2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5)]
√400 = √[(2 ⋅ 2)(2 ⋅ 2)(5 ⋅ 5)]
= 2 ⋅ 2 ⋅ 5
= 20
Further,
2(200) = 400 = 202
Example 2 :
Find the least number multiplied by 252 to get a perfect square.
Solution :
Decompose 252 into its prime factors.
Prime factors of 252 :
252 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7
= (2 ⋅ 2) ⋅ (3 ⋅ 3) ⋅ 7
The prime factor 7 does not occur in pair.
So, '7' is the least number to be multiplied by 252 to get a perfect square.
Justification :
√[7(252)] = √[7(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7)]
√1764 = √[(2 ⋅ 2)(3 ⋅ 3)(7 ⋅ 7)]
= 2 ⋅ 3 ⋅ 7
= 42
Further,
7(252) = 1764 = 422
Example 3 :
Find the least number multiplied by 180 to get a perfect square.
Solution :
Decompose 180 into its prime factors.
Prime factors of 180 :
180 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5
= (2 ⋅ 2) ⋅ (3 ⋅ 3) ⋅ 5
The prime factor 5 does not occur in pair.
So, '5' is the least number to be multiplied by 180 to get a perfect square.
Justification :
√[5(180)] = √[5(2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5)]
√900 = √[(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]
= 2 ⋅ 3 ⋅ 5
= 30
Further,
5(180) = 900 = 302
Example 4 :
Find the least number multiplied by 90 to get a perfect square.
Solution :
Decompose 90 into its prime factors.
Prime factors of 90 :
90 = 2 ⋅ 3 ⋅ 3 ⋅ 5
= 2 ⋅ (3 ⋅ 3) ⋅ 5
The prime factors 2 and 5 do not occur in pair.
Product of 2 and 5 :
2 ⋅ 5 = 10
So, '10' is the least number to be multiplied by 90 to get a perfect square.
Justification :
√[10(90)] = √[10(2 ⋅ 3 ⋅ 3 ⋅ 5)]
√900 = √[(2 ⋅ 5)(2 ⋅ 3 ⋅ 3 ⋅ 5)]
= √[(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]
= 2 ⋅ 3 ⋅ 5
= 30
Further,
10(90) = 900 = 302
Example 5 :
Find the least number multiplied by 120 to get a perfect square.
Solution :
Decompose 120 into its prime factors.
Prime factors of 120 :
120 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5
= (2 ⋅ 2) ⋅ 2 ⋅ 3 ⋅ 5
The prime factors 2, 3 and 5 do not occur in pair.
Product of 2, 3 and 5 :
2 ⋅ 3 ⋅ 5 = 30
So, '30' is the least number to be multiplied by 120 to get a perfect square.
Justification :
√[30(120)] = √[30(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)]
√3600 = √[(2 ⋅ 3 ⋅ 5)(2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5)]
= √[(2 ⋅ 2)(2 ⋅ 2)(3 ⋅ 3)(5 ⋅ 5)]
= 2 ⋅ 2 ⋅ 3 ⋅ 5
= 60
Further,
30(120) = 3600 = 602
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