Solve (3²×3)²÷√3⁶: A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving deep into a fascinating mathematical problem: (3²×3)²÷√3⁶. This looks a bit intimidating at first glance, but don't worry, we're going to break it down step by step, making it super easy to understand. We'll explore the order of operations, exponent rules, and how to handle square roots. So grab your calculators (or your brainpower!) and let's get started!
Understanding the Basics: Order of Operations
Before we even think about diving into the exponents and square roots, it's crucial to remember our old friend, the order of operations, often remembered by the acronym PEMDAS (or BODMAS in some parts of the world). This is our roadmap for solving any mathematical expression, ensuring we tackle the operations in the correct sequence.
- Parentheses (or Brackets)
- Exponents (or Orders)
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
This order is absolutely critical. If we don't follow it, we'll end up with the wrong answer. Think of it like following a recipe – if you add the ingredients in the wrong order, your cake might not turn out so great! So, let's keep PEMDAS (or BODMAS) in mind as we unravel our problem.
In our specific problem, (3²×3)²÷√3⁶, we see parentheses, exponents, multiplication, division, and a square root (which is essentially another form of exponent). This means we've got a good mix of operations to keep things interesting. Our first mission? To conquer those parentheses!
Tackling the Parentheses: Simplifying (3²×3)
The first thing we need to address according to PEMDAS is what's inside the parentheses: (3²×3). Now, we're dealing with exponents and multiplication here. Remember, exponents tell us how many times to multiply a number by itself. So, 3² means 3 multiplied by itself, which is 3 * 3 = 9. Great! Now we can rewrite the expression inside the parentheses as (9 × 3).
But wait, there's more! We still have that multiplication to take care of. 9 multiplied by 3 is a straightforward 27. So, we've successfully simplified the expression inside the parentheses: (3²×3) = 27. Feels good to knock out the first step, right? We're making progress already!
Now, let’s think about why this step is so important. Guys, imagine if we decided to multiply 3 by 3 first, then square the result. We'd get a completely different answer! That's why the order of operations is our mathematical GPS, guiding us to the correct destination. So, with the parentheses conquered, our original problem now looks a bit simpler: 27²÷√3⁶. We're one step closer to the final solution!
Exponents and Square Roots: Unveiling the Next Layer
Okay, with the parentheses out of the way, we're ready to tackle the next layer of our mathematical onion: exponents and square roots. Our expression now looks like this: 27²÷√3⁶. We've got a square (an exponent of 2) and a square root to deal with. Let's break it down.
First up, let's handle the exponent: 27². This means 27 multiplied by itself, which is 27 * 27. If you punch that into your calculator (or do the long multiplication), you'll find that 27² = 729. Fantastic! We've just simplified another part of our problem. Things are really starting to come together.
Now, let's turn our attention to the square root: √3⁶. This might look a bit trickier, but don't sweat it. Remember that a square root is the inverse operation of squaring a number. It's asking:
