Hey there, math enthusiasts! Ever get that itch to solve a good numerical puzzle? Well, today we're diving headfirst into a fascinating comparison of mathematical expressions. We'll break down each one, step-by-step, to unveil which calculation truly holds the crown. Get your thinking caps on, and let's get started!
Unraveling the Numerical Showdown
So, our mission, should we choose to accept it (and of course, we do!), is to determine which of these mathematical expressions yields the greatest result. We have a lineup of calculations ranging from multiplication and division to addition, some with negative numbers thrown in for an extra twist. Don't worry, guys, we'll tackle each one methodically, making sure no number is left behind. Let's jump right into our first contender!
(i) The Multiplication Marathon: $(-4) \times(-22) \times 4 \times(-3)$
Okay, let's get started with this multiplication marathon. In this mathematical expression, multiplication is the key. We have a series of numbers being multiplied together, including some negative ones. Remember, the golden rule here is that a negative times a negative equals a positive, and a positive times a negative equals a negative. So, let's break it down:
First, let's multiply the first two numbers: (-4) x (-22). A negative times a negative gives us a positive, so we have:
(-4) x (-22) = 88
Now, let's bring in the next number, which is 4. We multiply 88 by 4:
88 x 4 = 352
Finally, we multiply our result by the last number, (-3):
352 x (-3) = -1056
So, after all the multiplying, the result of our first expression is -1056. Keep that number in mind as we move on to our next contender.
Remember, the order of operations is super important here. We tackled the multiplication from left to right, ensuring we didn't miss any negative signs along the way. It's like a mathematical relay race, where each number passes the baton to the next, and we carefully calculate the result at each stage. The negative signs are like little hurdles, but we've cleared them with ease! Now, let's see what the next expression has in store for us.
(ii) Addition and Multiplication Mix: $7 + 6 \times(-4)$
Alright, next up, we have a mix of addition and multiplication. This is where the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), becomes super important. Guys, remember, multiplication comes before addition!
So, in the expression 7 + 6 x (-4), we need to tackle the multiplication first:
6 x (-4) = -24
Now that we've handled the multiplication, we can move on to the addition:
7 + (-24) = 7 - 24 = -17
Therefore, the result of this expression is -17. It's a significantly different number than our first result, isn't it? This highlights the power of the order of operations. If we had added 7 and 6 before multiplying, we would have gotten a completely different (and incorrect) answer. It's like following a recipe – you need to add the ingredients in the right order to get the delicious dish you're aiming for. In this case, the dish is the correct answer, and we've nailed it!
Consider this, the negative sign played a crucial role here, transforming a positive multiplication into a negative one, which ultimately affected the final result. This is a key takeaway when dealing with mathematical expressions: pay close attention to those signs! They can be game-changers. Now, let's move on to our third expression and see what mathematical magic it holds.
(iii) Division Duel: $-49 \div 7$ or $-49 \div(-7)$
Now we're entering the division arena! We have not one, but two division problems to solve here. This is like a mini-tournament within our larger competition. Let's take each division problem one at a time and see what results they yield.
First up, we have -49 ÷ 7. Remember the rules for dividing with negative numbers: a negative divided by a positive is a negative. So:
-49 ÷ 7 = -7
Now, let's tackle the second division problem: -49 ÷ (-7). Here, we have a negative divided by a negative, which results in a positive:
-49 ÷ (-7) = 7
So, we have two results: -7 and 7. Notice the difference a single negative sign can make! In the first case, the negative sign in the dividend (the number being divided) resulted in a negative quotient (the answer). In the second case, the negative sign in both the dividend and the divisor (the number we're dividing by) canceled each other out, giving us a positive quotient. It's like the negative signs are little ninjas, sometimes working against us and sometimes working in our favor. Keep that in mind.
This division duel highlights the importance of understanding the rules of signs in mathematical operations. Division, like multiplication, is heavily influenced by these rules. Getting them right is crucial for arriving at the correct answer. Now, with these two results in hand, let's move on to our final expression and complete our mathematical quest!
(iv) The Mixed Operations Mayhem: $(-9) \times 2 \div(-2) \times 7$ or $9 \times(-2) \div 2$
Okay, guys, brace yourselves! Our final contender is a bit of a mixed bag, featuring multiplication and division all in one expression. And, to make things even more interesting, we have two expressions to compare within this section. This is like the final boss level of our math game! Let's break it down step by step, remembering our trusty order of operations (PEMDAS) which, in this case, means we perform multiplication and division from left to right.
Let's start with the first expression: (-9) x 2 ÷ (-2) x 7
First, we tackle the multiplication: (-9) x 2 = -18
Next, we move on to the division: -18 ÷ (-2). A negative divided by a negative is a positive, so: -18 ÷ (-2) = 9
Finally, we multiply by 7: 9 x 7 = 63
So, the result of the first expression is 63.
Now, let's conquer the second expression: 9 x (-2) ÷ 2
First, we multiply: 9 x (-2) = -18
Then, we divide: -18 ÷ 2 = -9
So, the result of the second expression is -9.
Notice the contrast? The first expression, with its strategic negative signs, ultimately yielded a positive result, while the second expression ended up in the negative territory. This showcases how the placement and combination of operations can dramatically alter the outcome. It's like a mathematical dance, where each step (operation) influences the next, leading to a unique final pose (result).
This final expression really puts our understanding of mixed operations and the rules of signs to the test. We've navigated the multiplication, division, and negative signs like seasoned math pros. Now, with all our results calculated, it's time for the grand finale: determining which expression reigns supreme!
The Grand Reveal: Which Number Takes the Crown?
Alright, drumroll please! We've crunched the numbers, battled the negative signs, and conquered the order of operations. Now, it's time to reveal which of our mathematical expressions holds the title of