Simplify This Math Expression: (4-6)^2 + (10-6)^2 + (5-6)^2 / 5

Hello math enthusiasts! Today, we're diving into a fun arithmetic problem that will test your order of operations skills. We'll be breaking down and solving the expression: (4 minus 6) squared + (10 minus 6) squared + (5 minus 6) squared over 5. This might look a little intimidating at first glance, but with a step-by-step approach, we'll unravel it together. Let's get started!

Understanding the Expression

Our primary goal is to evaluate the mathematical expression: $(4 - 6)^2 + (10 - 6)^2 + (5 - 6)^2 / 5$. To do this accurately, we must follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This ensures that we perform calculations in the correct sequence to arrive at the single, correct answer.

The expression involves several operations: subtraction within parentheses, squaring (exponents), addition, and division. We'll tackle each part systematically. First, we address the operations inside the parentheses. This is crucial because anything within parentheses must be calculated before applying exponents or moving on to other operations outside them. It’s like peeling an onion, starting with the innermost layer.

Let's look at each parenthetical term:

1. (4 - 6): This is a simple subtraction. Four minus six equals negative two.
2. (10 - 6): Ten minus six equals four.
3. (5 - 6): Five minus six equals negative one.

After resolving the subtractions within the parentheses, our expression transforms into: $(-2)^2 + (4)^2 + (-1)^2 / 5$. This simplified form sets the stage for the next step: dealing with the exponents.

Applying the Exponents

Now that we've handled the parentheses, the next step in PEMDAS/BODMAS is to tackle the exponents, which are the 'squared' terms in our expression. Exponents indicate that we should multiply a base number by itself a certain number of times. In this case, all our exponents are '2', meaning we need to square each of the results from our parenthetical calculations.

Let's compute each squared term:

1. $(-2)^2$: This means multiplying -2 by itself. $(-2) \times (-2)$. Remember that a negative number multiplied by a negative number results in a positive number. So, $(-2)^2 = 4$.
2. $(4)^2$: This means multiplying 4 by itself. $4 \times 4 = 16$.
3. $(-1)^2$: This means multiplying -1 by itself. $(-1) \times (-1)$. Again, a negative times a negative is a positive. So, $(-1)^2 = 1$.

Substituting these squared values back into our expression, we get: $4 + 16 + 1 / 5$. We're getting closer to the final answer, and the expression is looking much simpler now.

Performing Division and Addition

We've successfully navigated the parentheses and exponents. The next part of PEMDAS/BODMAS involves Multiplication and Division, which are performed from left to right. In our current expression, $4 + 16 + 1 / 5$, we have a division operation: $1 / 5$. We must perform this before any addition.

1. $1 / 5$: Dividing 1 by 5 gives us 0.2. Alternatively, as a fraction, it remains $1/5$.

Now, our expression is: $4 + 16 + 0.2$. The final step according to PEMDAS/BODMAS is Addition and Subtraction, performed from left to right. We have only addition left.

1. $4 + 16$: Adding these two numbers gives us 20.
2. $20 + 0.2$: Finally, adding 0.2 to 20 results in 20.2.

So, the complete evaluation of the expression $(4 - 6)^2 + (10 - 6)^2 + (5 - 6)^2 / 5$ is 20.2.

Alternatively, if we keep the result as a fraction:

Our expression after squaring was $4 + 16 + 1 / 5$. Combining the whole numbers first: $4 + 16 = 20$. So, we have $20 + 1/5$. To add these, we can express 20 as a fraction with a denominator of 5: $20 = 100/5$. Then, $100/5 + 1/5 = 101/5$. Both 20.2 and 101/5 are correct representations of the final answer.

This exercise highlights the importance of following the order of operations meticulously. Each step builds upon the previous one, and a single misstep can lead to an entirely different result. Practicing such problems can significantly improve your mathematical fluency and confidence.

Conclusion

In summary, we successfully evaluated the expression $(4 - 6)^2 + (10 - 6)^2 + (5 - 6)^2 / 5$ by carefully following the order of operations (PEMDAS/BODMAS). We first resolved the subtractions within the parentheses, then calculated the exponents, performed the division, and finally completed the additions. The final result is 20.2 (or 101/5 as a fraction). Keep practicing your math skills; the more you solve, the more comfortable you'll become with these calculations!

For further exploration into mathematical operations and order of operations, you can visit resources like Khan Academy or Math is Fun.