By what number 5000 should be multiplied to become a perfect square and find the square root of result number
The Correct Answer and Explanation is:
To find the number by which 5000 should be multiplied to become a perfect square, let’s go through the steps systematically.
Step 1: Prime Factorization of 5000
Start by performing the prime factorization of 5000: 5000÷2=2500(divideby2)5000 \div 2 = 2500 \quad (divide by 2)5000÷2=2500(divideby2) 2500÷2=1250(divideby2)2500 \div 2 = 1250 \quad (divide by 2)2500÷2=1250(divideby2) 1250÷2=625(divideby2)1250 \div 2 = 625 \quad (divide by 2)1250÷2=625(divideby2)
Now divide by 5: 625÷5=125625 \div 5 = 125625÷5=125 125÷5=25125 \div 5 = 25125÷5=25 25÷5=525 \div 5 = 525÷5=5 5÷5=15 \div 5 = 15÷5=1
So, the prime factorization of 5000 is: 5000=23×545000 = 2^3 \times 5^45000=23×54
Step 2: Analyze the Exponents
To make 5000 a perfect square, each exponent in its prime factorization must be even. Currently, the exponents are:
- 232^323 (odd exponent),
- 545^454 (even exponent).
To make the number a perfect square, we need to balance the odd exponent of 2. Specifically, we need to multiply by one more factor of 2 to make the exponent of 2 even (i.e., 242^424).
Step 3: Multiply 5000 by 2
Thus, we need to multiply 5000 by 2: 5000×2=100005000 \times 2 = 100005000×2=10000
Step 4: Find the Square Root of 10000
Now, to find the square root of the result, 10000\sqrt{10000}10000: 10000=100\sqrt{10000} = 10010000=100
Final Answer:
- The number that 5000 should be multiplied by is 2.
- The square root of 10000 is 100.
Explanation:
By multiplying 5000 by 2, we ensure that each prime factor’s exponent becomes even, making the number a perfect square. The square root of 10000 is then the integer 100, which is the final answer
