Solving the Square Root of Fractions and Decimals

Square roots and fractions often require careful attention to detail. In this article, we will explore the process of finding the square root of a mixed fraction, specifically 2 and 1/4, using multiple methods. We will also discuss the importance of clarity in mathematical notation and provide step-by-step solutions.

Clarity in Mathematical Notation

Notational errors can lead to confusion. For instance, the fraction 2 and 1/4 can be ambiguous without clear notation. When written out in full, 2 and 1/4 means 9/4, while 2 of a quarter means 2 times 1/4, which is 1/2. Therefore, it's crucial to be explicit when expressing fractions and numbers. If the intended number is 9/4, then it should be written as '2 and a quarter,' or 9/4, to avoid any misinterpretations.

Methods for Solving the Square Root of 2 and 1/4

We will now provide two methods to find the square root of 2 and 1/4, also known as 9/4.

Method 1: Fractional Representation

( sqrt{frac{9}{4}} frac{sqrt{9}}{sqrt{4}} frac{3}{2} )

This method leverages the property that the square root of a fraction is the square root of the numerator divided by the square root of the denominator. Hence, ( sqrt{frac{9}{4}} frac{sqrt{9}}{sqrt{4}} frac{3}{2} ). Converting this to a mixed number, we have 1 and 1/2.

Method 2: Decimal Conversion

First, we convert 2 and 1/4 to its decimal form, which is 2.25. Then we find the square root of 2.25.

Using long division or a calculator, we find that:

2.25 / 1 2.25, 2.25 / 1.625 1.3846, 2.25 / 1.5048 1.4952, and finally, 2.25 / 1.5 1.5.

Hence, ( sqrt{2.25} 1.5 ).

Verification

To verify, we can square 1.5:

( 1.5 times 1.5 2.25 )

This confirms that the square root of 2.25 is indeed 1.5.

Adjusted Solution for Misinterpretations

Given the potential misinterpretations, we will also address two possible misreadings:

Case 1: (sqrt{21/4})

( sqrt{frac{21}{4}} pm frac{sqrt{21}}{2} )

In this case, the square root of 21/4 is ±(sqrt(21)/2).

Case 2: (sqrt{2 1/4})

( sqrt{2 frac{1}{4}} sqrt{frac{9}{4}} pm frac{3}{2} )

Here, the square root of 2 and 1/4 is ±3/2.

Conclusion

The square root of 2 and 1/4, or 2.25 in decimal form, is 1.5. Whether you prefer to work with fractions or decimals, the key is understanding the notation and applying the correct mathematical principles.