For each of the following numbers find the numbers find the smallest natural number by which it should be multiple so as to get a perfect square also find the square root of the perfect square so obtained  891

To solve this problem, we need to find the smallest natural number by which \( 891 \) should be multiplied to make it a perfect square. Then, we will find the square root of the perfect square obtained.

### Step 1: Prime Factorization of 891
First, we find the prime factorization of 891.

\[ 891 \div 3 = 297 \]
\[ 297 \div 3 = 99 \]
\[ 99 \div 3 = 33 \]
\[ 33 \div 3 = 11 \]
\[ 11 \div 11 = 1 \]

So, the prime factorization of 891 is:
\[ 891 = 3^4 \times 11^1 \]

### Step 2: Identify Missing Factors for a Perfect Square
For a number to be a perfect square, all the exponents in its prime factorization must be even. In the prime factorization of \( 891 \), \( 3^4 \) already has an even exponent, but \( 11^1 \) has an odd exponent.

To make \( 891 \) a perfect square, we need to multiply it by \( 11 \) (because \( 11^1 \times 11^1 = 11^2 \), making the exponent of 11 even).

### Step 3: Multiply and Find the Perfect Square
Multiply \( 891 \) by \( 11 \):

\[ 891 \times 11 = 9801 \]

### Step 4: Find the Square Root of the Perfect Square
Now, find the square root of \( 9801 \):

\[ \sqrt{9801} = 99 \]

### Final Answer
- **Smallest natural number to multiply**: \( 11 \)
- **Perfect square obtained**: \( 9801 \)
- **Square root of the perfect square**: \( 99 \)

Would you like any details or have any questions about this process? 

Here are some related problems you can explore:
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3. Calculate the square root of the perfect square obtained by multiplying 756 by the smallest number that makes it a perfect square.
4. If 500 must be multiplied by the smallest number to make it a perfect square, what is that number and the resulting square root?
5. Find the smallest number by which 1500 must be multiplied to become a perfect square and then determine its square root.

**Tip:** Always start by performing prime factorization when working with perfect squares.

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 891

Discover how to find the smallest natural number by which 891 should be multiplied to become a perfect square, and learn the process to find its square root. This problem explores prime factorization and properties of perfect squares, suitable for students in Grades 6-8.

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