Solving Complex Expressions: Simplifying (6^2 - 4^2)/5

Understanding complex expressions can often be overwhelming, especially when multiple operations like indices and division are involved. In this article, we will focus on the expression (6^2 - 4^2)/5, exploring different methods to simplify it and reach the final answer efficiently. By the end, you will have a clear understanding of the mathematical principles at play and how to handle similar expressions with ease.

Solving the Expression

Method 1: Direct Simplification

One of the more straightforward methods to solve the expression is by directly breaking it down into its components. Let's start by calculating the indices first.

$$ 6^2 36 4^2 16 $$

Substitute these values back into the original expression:

$$ frac{36 - 16}{5} frac{20}{5} 4 $$

Therefore, the final answer is:

boxed{4}

Method 2: Using the Difference of Squares

Another method to solve this expression is by using the difference of squares formula, which states that x^2 - y^2 (x y)(x - y). In our case, let:

$$ x 6, y 4 $$

Thus:

$$ 6^2 - 4^2 (6 4)(6 - 4) 10 cdot 2 20 $$

Now, solve the division:

$$ frac{20}{5} 4 $$

Therefore, the final answer is:

boxed{4}

Understanding BIDMAS/BODMAS

When dealing with complex expressions, understanding the order of operations (BIDMAS/BODMAS) is crucial. BIDMAS stands for Brackets, Indices, Division/Multiplication, Addition/Subtraction, and BODMAS stands for Brackets, Orders, Division/Multiplication, Addition/Subtraction. In our case, BODMAS indicates that indices take precedence over division.

Following BODMAS, the steps are:

B - Brackets: No brackets in the expression. I - Indices: Calculate the indices first: 6^2 36 4^2 16 D - Division and M - Multiplication: Once indices are calculated, solve the division: frac{36 - 16}{5} frac{20}{5} 4

Again, the final answer is:

boxed{4}

Other Simplified Methods

Some expressions can be computed more efficiently by recognizing patterns. For example:

$$6 - 4cdot64/5 2cdot10/2 2cdot2 4$$ $$6^2 - 4^2/5 36 - 16/5 20/5 4$$ $$(6^2 - 4^2)/5 (6 4)(6-4)/5 210/5 20/5 4$$

These methods are especially useful when the numbers are easy to work with and can be manipulated using simpler arithmetic operations.

Conclusion

In conclusion, the expression (6^2 - 4^2)/5 can be simplified using various methods, each with its own advantages. Whether you choose direct simplification, the difference of squares formula, or pattern recognition, the result remains the same: the final answer is 4.

Understanding these methods not only helps in solving similar expressions quickly but also enhances your problem-solving skills in mathematics. Whether you are a student, a teacher, or someone who enjoys math, these techniques will be valuable in your mathematical journey.