Equivalent Expression: $486-9+6+3 \times 2$ Solved

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Hey everyone! Let's break down this mathematical expression and figure out which option is the real deal. We've got a bit of a puzzle here: $486-9+6+3 \times 2$. The goal is to find an equivalent expression from the choices given. But before we jump into the options, let's tackle the original expression head-on. We need to remember our order of operations, the famous PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This is our golden rule in the math world!

Decoding the Original Expression

So, following PEMDAS, we first look for any parentheses. Nope, nothing there! Next up are exponents, and again, we don't have any. Now we move onto the big guns: multiplication and division. We spot a $3 \times 2$ lurking at the end. Let's handle that first.

$3 \times 2 = 6$

Now our expression looks like this: $486 - 9 + 6 + 6$. Much simpler, right? We've tamed the multiplication beast. The next step is to handle addition and subtraction. Remember, these are like the cool twins of the math world – they have equal power and we solve them from left to right. So, let's start from the left and work our way across the expression.

First up, we have $486 - 9$. Let's do some quick subtraction:

$486 - 9 = 477$

Great! Now our expression is even sleeker: $477 + 6 + 6$. Let's keep going from left to right. Next, we add 6 to 477:

$477 + 6 = 483$

We're almost there! Now we have $483 + 6$. Just one more addition to conquer:

$483 + 6 = 489$

Boom! We've cracked the code. The original expression, $486 - 9 + 6 + 3 \times 2$, equals 489. This is our target. Now, the real fun begins – we need to see which of the options matches this value. It's like a mathematical treasure hunt!

Option A: $\left[(486-9)+\left(6+3^3\right)\right] \times 2$

Alright, let's dive into Option A: $\left[(486-9)+\left(6+3^3\right)\right] \times 2$. This one looks a bit more complex, but don't worry, we'll tackle it step by step, just like before. Remember PEMDAS? It's our trusty guide. First up, we have parentheses galore! We need to work inside them first.

Let's start with the innermost parentheses: $(486 - 9)$. We already calculated this in our original expression breakdown, but let's do it again for good measure:

$486 - 9 = 477$

Okay, that's one set of parentheses down. Now let's look at the other set: $(6 + 3^3)$. Ooh, we've got an exponent sighting! According to PEMDAS, exponents come before addition, so let's handle that $3^3$ first.

$3^3$ means 3 multiplied by itself three times: $3 \times 3 \times 3 = 27$

Now we can substitute that back into our parentheses: $(6 + 27)$. Let's add those up:

$6 + 27 = 33$

Fantastic! We've conquered the exponents and the innermost parentheses. Now our expression looks like this: $[477 + 33] \times 2$. We still have those outer brackets to deal with, so let's add 477 and 33:

$477 + 33 = 510$

Now we're down to the wire! Our expression is a lean, mean $510 \times 2$. Let's multiply that out:

$510 \times 2 = 1020$

Whoa! Option A clocks in at 1020. That's quite a bit different from our target of 489. So, Option A is definitely not the equivalent expression we're looking for. But hey, we gave it a good shot, and we're getting closer to the answer. Onward to the next option!

Option B: $486-\left[\left(9+6+3^3\right) \times 2\right]$

Alright, let's set our sights on Option B: $486-\left[\left(9+6+3^3\right) \times 2\right]$. This one has a bit of a different structure, but we're not intimidated! We'll use our trusty PEMDAS compass to guide us through. Just like before, we need to tackle the parentheses first. This time, we have nested parentheses, so we'll start with the innermost ones: $\left(9+6+3^3\right)$.

Inside these parentheses, we have addition and an exponent. Remember, exponents take precedence, so let's calculate $3^3$ first:

$3^3 = 3 \times 3 \times 3 = 27$

Now we can replace $3^3$ with 27 in our innermost parentheses: $(9 + 6 + 27)$. Let's add those numbers together:

$9 + 6 + 27 = 42$

Great! We've conquered the innermost parentheses. Now our expression looks like this: $486 - [42 \times 2]$. We still have those square brackets to deal with, and inside them, we have a multiplication. Let's take care of that:

$42 \times 2 = 84$

Now our expression is streamlined: $486 - 84$. This looks much more manageable! Let's subtract 84 from 486:

$486 - 84 = 402$

Hmm, Option B gives us 402. That's not our target of 489 either. We're learning what doesn't work, which is still super valuable! Two options down, two to go. Let's keep our detective hats on and move on to Option C.

Option C: $(486-9)+6\left(3^3 \times 2\right)$

Okay, team, let's tackle Option C: $(486-9)+6\left(3^3 \times 2\right)$. This one's got a slightly different vibe, but we're not backing down! We know the drill – PEMDAS is our guide. First up, parentheses! We've got $(486 - 9)$ at the beginning. We've seen this before, but let's recalculate to be sure:

$486 - 9 = 477$

Awesome! Now, let's peek inside the other set of parentheses: $\left(3^3 \times 2\right)$. We've got an exponent and a multiplication hanging out together. Exponents come first, so let's handle that $3^3$:

$3^3 = 3 \times 3 \times 3 = 27$

Now we can replace $3^3$ with 27 in our parentheses: $(27 \times 2)$. Let's multiply those bad boys:

$27 \times 2 = 54$

Fantastic! We've conquered the parentheses. Now our expression looks like this: $477 + 6(54)$. Notice that the 6 is right next to the parentheses, which means multiplication is in the air! Let's multiply 6 by 54:

$6 \times 54 = 324$

Now our expression is taking shape: $477 + 324$. Just one more addition to go! Let's add 477 and 324:

$477 + 324 = 801$

Whoa, Option C gives us a whopping 801! That's way off from our target of 489. So, Option C is not our equivalent expression. We've eliminated three options, which means there's only one left... But we're not going to just assume it's the right answer. Let's put Option D through its paces and make sure it truly matches our target.

Option D: $(486-9+6)+\left(3^3 \times 2\right)$

Alright, let's give Option D a whirl: $(486-9+6)+\left(3^3 \times 2\right)$. This is our last contender, but we're going to treat it with the same rigorous approach we've used for the others. PEMDAS, here we come! We've got parentheses on both sides of the plus sign, so let's tackle them one at a time.

First up, let's look at $(486 - 9 + 6)$. This is a mix of subtraction and addition, so we'll work from left to right. We've actually seen $486 - 9$ before, but let's do it again:

$486 - 9 = 477$

Now we have $477 + 6$. Let's add those up:

$477 + 6 = 483$

Okay, the first set of parentheses is tamed! Now let's move on to the second set: $\left(3^3 \times 2\right)$. We've got an exponent and a multiplication. Exponents come first, so let's handle $3^3$:

$3^3 = 3 \times 3 \times 3 = 27$

Now we replace $3^3$ with 27: $(27 \times 2)$. Let's multiply those numbers:

$27 \times 2 = 54$

Fantastic! Both sets of parentheses are conquered. Our expression now looks like this: $483 + 54$. This is the home stretch! Let's add 483 and 54:

$483 + 54 = 489$

Yes! Option D gives us 489, which is exactly the value of our original expression. We've found our equivalent expression! It's been a mathematical marathon, but we've crossed the finish line.

Conclusion: The Winning Expression

After a thorough investigation, we've discovered that Option D, $(486-9+6)+\left(3^3 \times 2\right)$, is the expression equivalent to $486-9+6+3 \times 2$. We used PEMDAS as our guiding star, broke down each option step by step, and emerged victorious. Math can be a bit like a puzzle sometimes, but with a systematic approach and a dash of perseverance, we can crack any code! Great job, everyone!