Solving $1.25 \times 4 + 3 \times 2 \div (\frac{1}{2})^3$

Nov 5, 2025 by Admin 58 views

$Solving the Mathematical Expression: $1.25 \times 4 + 3 \times 2 \div (\frac{1}{2})^3$$

Hey guys! Today, we're diving into a fun mathematical problem. We're going to break down the expression $1.25 \times 4 + 3 \times 2 \div (\frac{1}{2})^3$ step by step, so you can see exactly how to solve it. Math can seem intimidating, but I promise, if we take it slowly and follow the order of operations, it’s totally manageable. So, grab your calculators (or your mental math muscles!) and let's get started!

Understanding the Order of Operations

Before we jump into the nitty-gritty calculations, let's quickly recap the order of operations, often remembered by the acronym PEMDAS (or BODMAS, depending on where you learned math!). This is super important because it tells us in what order we need to perform the different operations to get the correct answer. If we don't follow this order, we might end up with a completely different result, and nobody wants that!

Parentheses (or Brackets)
Exponents (or Orders)
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

So, first, we tackle anything inside parentheses or brackets. Then, we deal with exponents (those little superscript numbers). After that, we perform multiplication and division, making sure to work from left to right. Finally, we handle addition and subtraction, also from left to right. Keeping this order in mind is the key to unlocking the solution to our expression!

Breaking Down the Expression Step-by-Step

Let's apply PEMDAS to our expression: $1.25 \times 4 + 3 \times 2 \div (\frac{1}{2})^3$ .

Exponents: First up, we need to deal with the exponent. We have $(\frac{1}{2})^3$ , which means $\frac{1}{2}$ multiplied by itself three times: $(\frac{1}{2}) \times (\frac{1}{2}) \times (\frac{1}{2}) = \frac{1}{8}$ . So, we can rewrite our expression as: $1.25 \times 4 + 3 \times 2 \div \frac{1}{8}$ .
Multiplication and Division: Next, we handle multiplication and division, working from left to right. We have $1.25 \times 4$ , which equals 5. So, our expression now looks like: $5 + 3 \times 2 \div \frac{1}{8}$ . Moving along, we have $3 \times 2$ , which is 6. Our expression is now: $5 + 6 \div \frac{1}{8}$ . Now comes the division: $6 \div \frac{1}{8}$ . Remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{1}{8}$ is 8, so we have $6 \times 8 = 48$ . Our expression simplifies to: $5 + 48$ .
Addition: Finally, we perform the addition: $5 + 48 = 53$ .

So, after following the order of operations, we've arrived at the solution: $1.25 \times 4 + 3 \times 2 \div (\frac{1}{2})^3 = 53$ .

Diving Deeper into Each Operation

To really nail this, let's take a closer look at each of the operations we performed. Understanding the 'why' behind each step can make the whole process feel less like a set of rules and more like a logical puzzle.

Tackling Exponents: $(\frac{1}{2})^3$

Exponents can sometimes look a bit scary, but they're actually pretty straightforward. An exponent simply tells you how many times to multiply a number (the base) by itself. In our case, we had $(\frac{1}{2})^3$ . The base is $\frac{1}{2}$ , and the exponent is 3. This means we multiply $\frac{1}{2}$ by itself three times. When multiplying fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, $(\frac{1}{2}) \times (\frac{1}{2}) \times (\frac{1}{2}) = \frac{1 \times 1 \times 1}{2 \times 2 \times 2} = \frac{1}{8}$ . Understanding exponents is crucial, especially when dealing with more complex mathematical expressions and scientific notation.

Mastering Multiplication and Division: $1.25 \times 4$ and $6 \div \frac{1}{8}$

Multiplication and division are like two sides of the same coin. They're inverse operations, meaning one undoes the other. In our expression, we had $1.25 \times 4$ . This is a simple multiplication, resulting in 5. But then we encountered $6 \div \frac{1}{8}$ , which is a little trickier. Dividing by a fraction can seem confusing, but there's a neat trick: dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is what you get when you flip it over. So, the reciprocal of $\frac{1}{8}$ is $\frac{8}{1}$ , which is just 8. Therefore, $6 \div \frac{1}{8}$ is the same as $6 \times 8$ , which equals 48. This reciprocal trick is a powerful tool in your math arsenal!

The Grand Finale: Addition - $5 + 48$

Addition is often the most straightforward operation. In our final step, we had $5 + 48$ . Simply adding these two numbers together gives us 53. It’s the culmination of all the previous steps, bringing us to the final answer. Addition is a fundamental operation, and it's essential for building a strong foundation in mathematics.

Common Mistakes to Avoid

When tackling expressions like this, there are a few common pitfalls to watch out for. Avoiding these mistakes can save you a lot of headaches and ensure you get the correct answer.

Ignoring the Order of Operations

This is the biggest mistake people make! It’s super tempting to just work through the expression from left to right, but that will almost always lead to the wrong answer. Always, always, always remember PEMDAS (or BODMAS) – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Get this order ingrained in your brain, and you'll be well on your way to math mastery.

Miscalculating Exponents

It's easy to make a mistake with exponents, especially if you rush. Remember that an exponent tells you how many times to multiply the base by itself, not by the exponent. So, $(\frac{1}{2})^3$ is $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}$ , not $\frac{1}{2} \times 3$ . Take your time and double-check your calculations to avoid this error.

Forgetting the Reciprocal Rule

Dividing by a fraction can be confusing if you don't remember the reciprocal rule. Remember, dividing by a fraction is the same as multiplying by its reciprocal. Flip the fraction over, and then multiply. This simple trick can turn a tricky division problem into a much simpler multiplication problem.

Arithmetic Errors

Simple arithmetic errors can happen to anyone, especially when dealing with multiple operations. It's always a good idea to double-check your calculations, especially in longer expressions. A small mistake early on can throw off the entire solution.

Real-World Applications

You might be wondering,