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:

1. Find the smallest number by which 180 must be multiplied to get a perfect square.
2. Determine the smallest number by which 450 must be multiplied to get a perfect square.
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.