Solving The Math Expression: A Step-by-Step Guide

Hey guys! Let's break down this math expression together. It looks a bit intimidating at first, but trust me, we can handle it. We're going to take it step by step, making sure we understand each part before moving on. So, grab your calculators (or your mental math skills!), and let's dive in! Our focus will be on understanding the individual components of the expression and how they fit together. We'll be working with square roots, fractions, and basic arithmetic operations, so it’s a good review of some fundamental math concepts. Remember, the key to solving any complex problem is to break it down into smaller, more manageable parts. We'll tackle the square roots first, then the fractions, and finally combine everything using the correct order of operations. Math can be fun, especially when we approach it with a clear and organized method. Let’s get started and see how this expression unfolds! And remember, there's no such thing as a silly question. If something doesn’t make sense, ask! We're all here to learn and support each other. So, let’s put on our thinking caps and get ready to solve!

Breaking Down the Expression

Okay, first things first, let's write down the expression we're tackling: ${ \left(-6 \sqrt{\frac{1}{4}} + \frac{\sqrt{324}}{2}, \frac{\sqrt{0.16}}{2} \right) : \sqrt{25} - 2 }$

It might look like a jumble of symbols and numbers, but don't worry! We're going to dissect it piece by piece. The first thing we notice is the parentheses. Anything inside the parentheses needs to be simplified first. Inside the parentheses, we've got a mix of square roots and fractions. Remember, the square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. Fractions are just a way of representing parts of a whole. So, ${\frac{1}{4}}$ means one out of four parts. We also see a colon (:), which represents division. And, of course, we have subtraction (-) at the very end. To make things super clear, let's identify the different components we need to simplify: We have the square root of ${\frac{1}{4}}$, which is a fraction. Then there's the square root of 324, which is a larger number. We also have ${\sqrt{0.16}}$, which involves a decimal. Lastly, we have ${\sqrt{25}}$, a familiar square root. Our initial strategy is to simplify these square roots and fractions as much as possible before we start combining them. This will make the whole process much smoother and less prone to errors. Think of it like preparing all your ingredients before you start cooking – it just makes everything flow better!

Simplifying the Square Roots

Now, let's get down to business and simplify those square roots. This is where things start to get interesting! Remember, finding the square root of a number is like asking, "What number, when multiplied by itself, gives me this number?" Let's begin with the first square root we encounter: ${\sqrt{\frac{1}{4}}}$ This might look tricky because it's a fraction inside a square root, but it's actually quite straightforward. We can think of the square root applying to both the numerator (the top number) and the denominator (the bottom number) separately. So, we need to find the square root of 1 and the square root of 4. What number multiplied by itself equals 1? Well, that's easy – it's 1! And what number multiplied by itself equals 4? That's 2, because 2 * 2 = 4. Therefore, ${\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}}$ See? Not so scary after all! Next up, we have ${\sqrt{324}}$. This one might not be as obvious, but we can either use a calculator or try to think of perfect squares that might be factors of 324. A little trial and error, or a quick calculator check, reveals that 18 * 18 = 324. So, ${\sqrt{324} = 18}$. Moving on, let's tackle ${\sqrt{0.16}}$. This involves a decimal, but we can handle it. Think of 0.16 as the fraction ${\frac{16}{100}}$. Now we have ${\sqrt{\frac{16}{100}}}$ which we can again separate into ${\frac{\sqrt{16}}{\sqrt{100}}}$ The square root of 16 is 4 (4 * 4 = 16), and the square root of 100 is 10 (10 * 10 = 100). So, ${\sqrt{0.16} = \frac{4}{10} = 0.4}$ Finally, we have ${\sqrt{25}}$, which is a classic and easy one. 5 * 5 = 25, so ${\sqrt{25} = 5}$. We've now successfully simplified all the square roots in our expression. That's a huge step forward! We're making great progress. Now, let's substitute these simplified values back into our original expression and see what we've got.

Substituting and Simplifying Further

Alright, we've done the hard work of simplifying the square roots. Now comes the fun part – plugging those values back into our original expression and seeing how things shake out. It’s like putting the pieces of a puzzle together! Remember our expression? It was: ${ \left(-6 \sqrt{\frac{1}{4}} + \frac{\sqrt{324}}{2}, \frac{\sqrt{0.16}}{2} \right) : \sqrt{25} - 2 }$ We figured out that ${\sqrt{\frac{1}{4}} = \frac{1}{2}}$, ${\sqrt{324} = 18}$, ${\sqrt{0.16} = 0.4}$, and ${\sqrt{25} = 5}$. Let's substitute these values in: ${ \left(-6 \cdot \frac{1}{2} + \frac{18}{2}, \frac{0.4}{2} \right) : 5 - 2 }$ See how much cleaner it looks already? We've replaced those square roots with simple numbers. Now, let's focus on simplifying inside the parentheses first. This is where the order of operations (PEMDAS/BODMAS) comes into play. Remember, we do Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). Inside the first part of the parentheses, we have -6 multiplied by ${\frac{1}{2}}$, which is -3. Then we have ${\frac{18}{2}}$, which simplifies to 9. So, the first part inside the parentheses becomes -3 + 9. In the second part of the parentheses, we have ${\frac{0.4}{2}}$, which is 0.2. Now our expression looks like this: ${ \left(-3 + 9, 0.2 \right) : 5 - 2 }$ We can further simplify -3 + 9 to 6. So, our expression is now: ${ \left(6, 0.2 \right) : 5 - 2 }$ We're getting closer! Now, we need to deal with the division. Remember that colon (:) means division. So, we're dividing the entire expression inside the parentheses by 5. This means we divide both 6 and 0.2 by 5.

Final Steps and the Solution

Okay, we're in the home stretch now! We've simplified the square roots, substituted the values, and worked our way through the parentheses. We're left with: ${ \left(6, 0.2 \right) : 5 - 2 }$ As we discussed, the colon (:) signifies division. We need to divide both elements inside the parentheses by 5. So, let's do that: 6 divided by 5 is ${\frac{6}{5}}$, which is also equal to 1.2. And 0.2 divided by 5 is 0.04. Now our expression looks like this: ${ \left(1.2, 0.04 \right) - 2 }$ But wait! There's a slight misunderstanding in how the original expression was presented. The comma inside the parentheses suggests we're dealing with a coordinate pair or two separate expressions, not a single expression to be divided. My apologies for that oversight! Let’s revisit the original expression: ${ \left(-6 \sqrt{\frac{1}{4}} + \frac{\sqrt{324}}{2}, \frac{\sqrt{0.16}}{2} \right) : \sqrt{25} - 2 }$ Given the comma, it's more accurate to interpret this as two separate calculations, both then divided by ${\sqrt{25}}$, and finally, 2 subtracted from the result of the division. Let’s call the two parts A and B: ${ A = -6 \sqrt{\frac{1}{4}} + \frac{\sqrt{324}}{2} }$ ${ B = \frac{\sqrt{0.16}}{2} }$ We've already simplified these in the previous steps: ${ A = -6 \cdot \frac{1}{2} + \frac{18}{2} = -3 + 9 = 6 }$ ${ B = \frac{0.4}{2} = 0.2 }$ Now, we divide both A and B by ${\sqrt{25}}$ which is 5: ${ \frac{A}{5} = \frac{6}{5} = 1.2 }$ ${ \frac{B}{5} = \frac{0.2}{5} = 0.04 }$ Finally, we subtract 2 from the result. Since the expression is ambiguous, let's consider two possibilities:

1. Subtracting 2 from the result of A divided by 5: ${ \frac{A}{5} - 2 = 1.2 - 2 = -0.8 }$

2. Subtracting 2 from the result of B divided by 5: ${ \frac{B}{5} - 2 = 0.04 - 2 = -1.96 }$

So, depending on which part we subtract 2 from, we get -0.8 or -1.96. Given the ambiguity, it's crucial to clarify the intended order of operations. But based on our step-by-step breakdown and the most likely interpretation, these are the potential solutions. Wow, we really dug deep into this one! It's a fantastic example of how breaking down a complex problem into smaller steps can make it much more manageable. Remember, math is all about precision and paying attention to the details. And sometimes, it's about recognizing that there might be more than one way to interpret a problem. You guys did an awesome job sticking with it until the end! Give yourselves a pat on the back. You've earned it!