For each of the following number  find the smallest natural number by which it should be multiplied so as to get perfect square also find the square root of the perfect square so obtained 768

To solve this problem, we need to determine the smallest natural number by which 768 should be multiplied to obtain a perfect square. Then, we will find the square root of that perfect square.

### Step 1: Prime Factorization of 768
Let's first perform the prime factorization of 768.

- 768 ÷ 2 = 384
- 384 ÷ 2 = 192
- 192 ÷ 2 = 96
- 96 ÷ 2 = 48
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 768 is:

\[ 768 = 2^8 \times 3^1 \]

### Step 2: Determine the Smallest Number to Multiply
For 768 to become a perfect square, all the prime factors must have even exponents.

- The exponent of 2 is already even (8).
- The exponent of 3 is 1, which is odd.

To make the exponent of 3 even, we need to multiply 768 by 3. 

### Step 3: Multiply and Find the Square Root
Now, let's multiply 768 by 3:

\[ 768 \times 3 = 2304 \]

The new number, 2304, should be a perfect square.

### Step 4: Find the Square Root of 2304
The square root of 2304 is calculated as follows:

\[ \sqrt{2304} = 48 \]

### Final Answer
- **The smallest natural number to multiply 768 by to get a perfect square is 3.**
- **The square root of the resulting perfect square (2304) is 48.**

Would you like more details on any of these steps or have any questions?

Here are 5 related questions:
1. What is the prime factorization of a different number, say 900?
2. How do you determine if a number is already a perfect square?
3. What is the smallest natural number to multiply 675 to make it a perfect square?
4. How does multiplying by a prime factor ensure that a number becomes a perfect square?
5. What is the square root of a perfect square, and how is it different from the square root of a non-perfect square?

**Tip:** Always check the exponents in the prime factorization to see if any are odd, as this will tell you if a number needs to be multiplied to become a perfect square.

For each of the following numbers, find the smallest natural number by which it should be multiplied so as to get a perfect square; also find the square root of the perfect square so obtained. 768

Explore how to find the smallest natural number that multiplies 768 to form a perfect square and learn how to determine the square root of the resulting perfect square. This problem involves prime factorization and identifying conditions for perfect squares, suitable for Grades 6-8.

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