What is the Square Root of 20 1/4?

Introduction

Finding the square root of a fraction or a mixed number can be a practical skill, especially in various fields such as mathematics, engineering, and physics. This article delves into the process of finding the square root of 20 1/4, also written as 20.25, providing different methods and solutions for better understanding.

Method 1: Simplifying the Fraction

The first method involves converting the mixed fraction 20 1/4 into an improper fraction, then applying the square root to this fraction.

Step 1: Convert 20 1/4 to an improper fraction.

20 1/4 81/4

Step 2: Apply the square root to the fraction.

Square root of 81/4 √(81/4)

Step 3: Simplify the root.

√(81/4) √81 / √4 9 / 2

Step 4: Convert to a mixed fraction or decimal.

9/2 4 1/2 or 4.5

Method 2: Using Logarithms

This method utilizes logarithmic identities to simplify the calculation.

Step 1: Write 20.25 in terms of powers of 10.

20.25 ≈ 2^11 / 100

Step 2: Calculate the logarithm of 20.25.

log_{10}20.25 ≈ 1.31133

Step 3: Apply the square root to the logarithm.

log_{10}√20.25 ≈ 0.6557

Step 4: Convert back to the original number.

√20.25 ≈ 4.5

Method 3: Using the Binomial Theorem

This method demonstrates how to use the binomial theorem to simplify square root calculations.

Step 1: Write 20.25 as a sum of fractions.

20.25 ≈ (45/10)^2

Step 2: Simplify the fraction.

45/10 4.5

Step 3: Take the square root.

√20.25 4.5

Conclusion

In summary, the square root of 20 1/4, or 20.25, can be simplified and calculated using various methods. The most straightforward approach involves converting the mixed fraction into an improper fraction and then taking the square root. Alternatively, using logarithms and the binomial theorem can also lead to the correct result. Regardless of the method, the final answer is 4.5.

References

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