Simplifying the Square Root Expression

We need to evaluate the expression \(\sqrt{\frac{1.21\times0.9}{1.1\times0.11}}\). To simplify this, we can first simplify the fraction inside the square root.

Step 1: Simplify the Numerator

The numerator is the product of 1.21 and 0.9.

\[ 1.21 \times 0.9 \] Multiply the numbers without considering the decimal points first: \(121 \times 9 = 1089\).

Now, count the total number of decimal places in the numbers being multiplied. 1.21 has two decimal places and 0.9 has one decimal place. The total is \(2 + 1 = 3\) decimal places.

So, the product \(1.21 \times 0.9\) will have three decimal places. Starting from the right of 1089, move three places to the left to place the decimal point.

\[ 1.21 \times 0.9 = 1.089 \]

Step 2: Simplify the Denominator

The denominator is the product of 1.1 and 0.11.

\[ 1.1 \times 0.11 \] Multiply the numbers without considering the decimal points first: \(11 \times 11 = 121\).

Now, count the total number of decimal places. 1.1 has one decimal place and 0.11 has two decimal places. The total is \(1 + 2 = 3\) decimal places.

So, the product \(1.1 \times 0.11\) will have three decimal places. Starting from the right of 121, move three places to the left to place the decimal point.

\[ 1.1 \times 0.11 = 0.121 \]

Step 3: Simplify the Fraction Inside the Square Root

Now the expression inside the square root is \(\frac{1.089}{0.121}\).

To divide decimals, we can remove the decimal points by multiplying both the numerator and the denominator by a power of 10. Since both numbers have three decimal places, we multiply by \(10^3 = 1000\).

\[ \frac{1.089 \times 1000}{0.121 \times 1000} = \frac{1089}{121} \]

Now, we perform the division \(1089 \div 121\). We can recognize that \(121 = 11^2\). Let's try dividing 1089 by 121.

\[ 121 \times 9 \]

\[ 121 \times 9 = (100 + 20 + 1) \times 9 = 900 + 180 + 9 = 1089 \]

So, the division simplifies to:

\[ \frac{1089}{121} = 9 \]

Step 4: Calculate the Square Root

The expression inside the square root simplifies to 9. Now we need to find the square root of 9.

\[ \sqrt{9} \]

The square root of 9 is the number that, when multiplied by itself, equals 9. This number is 3.

\[ \sqrt{9} = 3 \]

Therefore, the value of the given expression is 3.

The final answer obtained after simplifying the expression is 3.