Find the smallest number by which 180 must be multiplied so it becomes a perfect square, and find the square root of the product obtained.

To make 180 a perfect square, it must be multiplied by 5, resulting in 900. The square root of 900 is 30. Thus, the required number is 5 and the square root is 30.
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Answered by Anonymous | 2025-07-07

Answers:
Smallest number to multiply with 180 would be ** 5 **
The square root of the product is ** 30 **
Explanation
Think of the list of perfect squares (1,4,9,16,25,36, etc)
Which of those is a factor of 180? Through trial-and-error you'll find that 36 is a factor of 180 and it's the largest such factor.
180/36 = 5 which means 180 = 36*5
The value 36 is a perfect square, but 5 is not. To transform that 5 into a perfect square, we multiply both sides by 5 like so:
180 = 36*5
180 5 = 36 5*5
180 5 = 36 25
The right hand side now has both parts as perfect squares. So the entire right hand side is also a perfect square (namely 36*25 = 900). Note how 30^2 = 900.
The square root of 900, aka 30^2, would be 30. The square root and squaring operation cancel each other out.

Answered by jimthompson5910 | 2025-07-07

To find the smallest number by which 180 must be multiplied so it becomes a perfect square, we need to factor 180 into its prime factors. Then, we can determine what is needed to make it a perfect square.

Prime Factorization of 180:
Let's break down 180 into its prime factors:
180 ÷ 2 = 90 90 ÷ 2 = 45 45 ÷ 3 = 15 15 ÷ 3 = 5 5 ÷ 5 = 1
So, the prime factorization of 180 is:
180 = 2 2 × 3 2 × 5

Determine Needed Multiplication Factor:
A perfect square needs to have an even power for all prime factors. Currently, we have 5 1 as the factor with an odd power. To make it even, we need one more factor of 5. Thus, the smallest number to multiply 180 by is 5.

Calculate the Product:
Multiply 180 by 5:
180 × 5 = 900

Calculate the Square Root of the Product:
Now, check if 900 is a perfect square by finding its square root:
900 ​ = 30
Thus, the smallest number by which 180 must be multiplied to make it a perfect square is 5, and the square root of the result, 900, is 30.


In summary, multiply 180 by 5 to get 900, a perfect square, with a square root of 30.

Answered by OliviaLunaGracy | 2025-07-08