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.