Simple Math: Solving An Algebraic Expression

by Alex Johnson 45 views

Ever looked at a mathematical expression and wondered how to approach it? Sometimes, the most straightforward problems can seem a bit daunting with all those numbers and operations. Today, we're going to break down a specific algebraic expression: (4 minus 6) squared + (10 minus 6) squared + (5 minus 6) squared over 5. Don't let the parentheses and exponents scare you; we'll walk through it step-by-step, making sure everything is crystal clear.

This problem tests our understanding of the order of operations, often remembered by the acronym PEMDAS or BODMAS. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS stands for Brackets, Orders, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Both acronyms emphasize the same hierarchy of operations, ensuring we solve mathematical expressions consistently and accurately.

Let's dissect the expression: (4−6)2+(10−6)2+(5−6)2/5(4-6)^2 + (10-6)^2 + (5-6)^2 / 5.

Our first step is to tackle the operations inside the parentheses. These are the innermost parts of the expression that need to be resolved before anything else.

  1. First Parenthesis: (4−6)(4-6) Subtracting 6 from 4 gives us -2. So, (4−6)=−2(4-6) = -2.

  2. Second Parenthesis: (10−6)(10-6) Subtracting 6 from 10 results in 4. So, (10−6)=4(10-6) = 4.

  3. Third Parenthesis: (5−6)(5-6) Subtracting 6 from 5 yields -1. So, (5−6)=−1(5-6) = -1.

Now, let's substitute these results back into our original expression. The expression transforms into:

(−2)2+(4)2+(−1)2/5(-2)^2 + (4)^2 + (-1)^2 / 5

The next operation according to PEMDAS/BODMAS is Exponents (or Orders). We need to square each of the numbers we just calculated.

  1. First Exponent: (−2)2(-2)^2 Squaring -2 means multiplying it by itself: (−2)×(−2)(-2) \times (-2). A negative number multiplied by a negative number results in a positive number. So, (−2)2=4(-2)^2 = 4.

  2. Second Exponent: (4)2(4)^2 Squaring 4 means multiplying it by itself: 4×44 \times 4. This gives us 16. So, (4)2=16(4)^2 = 16.

  3. Third Exponent: (−1)2(-1)^2 Squaring -1 means multiplying it by itself: (−1)×(−1)(-1) \times (-1). Again, a negative multiplied by a negative is positive. So, (−1)2=1(-1)^2 = 1.

Our expression now looks like this:

4+16+1/54 + 16 + 1 / 5

We're getting closer! The next step involves Multiplication and Division. In our current expression, we have a division operation: 1/51 / 5.

  1. Division: 1/51 / 5 Dividing 1 by 5 gives us 0.2. So, 1/5=0.21 / 5 = 0.2.

Now, the expression simplifies to:

4+16+0.24 + 16 + 0.2

Finally, we perform the Addition and Subtraction operations from left to right. In this case, we only have addition.

  1. First Addition: 4+164 + 16 Adding 4 and 16 gives us 20.

  2. Second Addition: 20+0.220 + 0.2 Adding 20 and 0.2 results in 20.2.

So, the final answer to the expression (4−6)2+(10−6)2+(5−6)2/5(4-6)^2 + (10-6)^2 + (5-6)^2 / 5 is 20.2.

This breakdown illustrates how the order of operations is crucial in solving mathematical problems. By systematically addressing parentheses, exponents, division, and finally addition, we arrived at the correct solution. It's a good reminder that even complex-looking expressions can be tackled with a clear, step-by-step approach.

Understanding the Components

Let's delve a little deeper into why each step is important and what mathematical principles are at play. The expression we solved, (4−6)2+(10−6)2+(5−6)2/5(4-6)^2 + (10-6)^2 + (5-6)^2 / 5, is a wonderful example of how different mathematical concepts interrelate.

Parentheses and Order of Operations: The parentheses are fundamental. They group operations, dictating which ones must be performed first. Without them, the expression could be interpreted in multiple ways, leading to different answers. The rule of