What is the sum of the square root of 4/9 and the square root of 9/16? Express your answer as a mixed number.

To solve this problem, you first need to tackle each fraction's square root individually. A helpful rule to remember is that the square root of a fraction is simply the square root of the numerator over the square root of the denominator. For the first fraction, the square root of 4 is 2 and the square root of 9 is 3, giving us 2/3. For the second, the square root of 9 is 3 and the square root of 16 is 4, resulting in 3/4. Both 4, 9, and 16 are "perfect squares," meaning their square roots are whole numbers, which simplifies the process.

The next step is to add the two resulting fractions: 2/3 and 3/4. To do this, we need a common denominator, which is the smallest number that both 3 and 4 can divide into evenly. In this case, that number is 12. We convert 2/3 into 8/12 (by multiplying the top and bottom by 4) and 3/4 into 9/12 (by multiplying the top and bottom by 3). Now, we can easily add them: 8/12 + 9/12 equals 17/12.

Finally, the answer must be expressed as a mixed number. The fraction 17/12 is an improper fraction because the numerator is larger than the denominator. To convert it, we see how many times 12 fits into 17. It fits one whole time with a remainder of 5. This gives us the final mixed number of 1 and 5/12. This type of calculation combines fundamental concepts of fractions and roots that have been used since ancient times, from Babylonian clay tablets to the construction of Egyptian pyramids.