Simplifying Mathematical Expressions: Correct Calculation of √75 - 3/√3

Simplifying Mathematical Expressions: Correct Calculation of √75 - 3/√3

When dealing with complex mathematical expressions, it's crucial to simplify them accurately to ensure clear and precise results. One such expression is √75 - 3/√3. This article will explore the correct calculation methods and provide a step-by-step simplification for this expression.

Step-by-Step Simplification

To simplify the expression √75 - 3/√3, we will break it down into manageable parts and follow a systematic approach. Here are the steps:

Method 1: Simplifying the Square Roots

The first step is to simplify the square roots. Notice that 75 can be factored as:

75 25 × 3

Therefore, √75 √(25 × 3) √25 × √3 5√3

The expression now becomes:

5√3 - 3/√3

Method 2: Rationalizing the Denominator

To eliminate the root in the denominator, we can multiply the numerator and the denominator of the fraction by the conjugate of the denominator:

(√3 - 1)

Multiplying the numerator and the denominator by √3 - 1, we get:

(5√3 - 3)(√3 - 1) / (3 - 1)

The denominator simplifies to 2:

(5√3 - 3)(√3 - 1) / 2

Expanding the numerator using the distributive property:

5√3 × √3 - 5√3 × 1 - 3 × √3 3 × 1

This simplifies to:

5 × 3 - 5√3 - 3√3 3

15 - 8√3 3

18 - 8√3

So the expression is now:

(18 - 8√3) / 2

Simplifying further:

9 - 4√3

Final Decimal Approximation

If you need the decimal approximation, using a pocket calculator can provide the exact value:

9 - 4√3 ≈ 2.071796…

Applications and Relevance

Understanding how to simplify and correctly calculate mathematical expressions is crucial in various fields, including:

Engineering

Physics

Finance

Computer Science

This skill enhances problem-solving abilities and ensures that calculations are accurate and reliable.

Conclusion

In conclusion, the correct simplification of the expression √75 - 3/√3 is accomplished through systematic steps involving the simplification of square roots and rationalization. The final expression is 9 - 4√3, which can be further approximated to 2.071796.... Practicing such simplifications will not only improve your mathematical skills but also provide a strong foundation for more complex calculations in various academic and professional fields.