Make Math Expressions Equal 10

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Hey guys! Ever stared at a math problem and wondered how to tweak it to get the answer you want? Today, we're diving into a fun little puzzle where we need to make the expression 4+42−5imes24+4^2-5 imes 2 equal 10. It's all about understanding the magic of order of operations and how parentheses can totally change the game. So, buckle up, because we're going to break this down step-by-step and figure out which modification, if any, will get us to our target number of 10.

Understanding the Original Expression

Before we start adding any fancy parentheses, let's figure out what the original expression, 4+42−5imes24+4^2-5 imes 2, evaluates to. This is super important because it gives us our starting point. Remember the order of operations, often remembered by the acronym PEMDAS or BODMAS? It stands for:

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

Let's apply this to our expression: 4+42−5imes24+4^2-5 imes 2.

  1. Exponents: First, we tackle 424^2. That's 4imes44 imes 4, which equals 16. So the expression becomes 4+16−5imes24+16-5 imes 2.
  2. Multiplication: Next, we do 5imes25 imes 2, which is 10. The expression now looks like 4+16−104+16-10.
  3. Addition and Subtraction: Now we work from left to right. First, 4+164+16 equals 20. Then, 20−1020-10 equals 10.

Wow! So, the original expression 4+42−5imes24+4^2-5 imes 2 actually already equals 10! This is a crucial piece of information, guys. It means that if the goal is to make the expression equal 10, then the answer is probably that we don't need to do anything. But let's explore the options just to be sure and to really solidify our understanding of how parentheses work.

Analyzing the Options: The Power of Parentheses

Now, let's look at the proposed changes and see how they affect the value of the expression. We're going to evaluate each option using PEMDAS to see if any of them also result in 10, or if they lead us astray.

Option A: Nothing needs to be done.

As we just discovered, the original expression 4+42−5imes24+4^2-5 imes 2 evaluates to 10. So, technically, this statement is correct. If the objective is to achieve a value of 10, and the expression already does that, then no changes are necessary. This is our baseline, and it's a strong contender for the right answer.

Option B: Add parentheses around 4+44+4.

If we add parentheses around 4+44+4, the expression becomes (4+4)+42−5imes2(4+4)+4^2-5 imes 2. Let's evaluate this:

  1. Parentheses: (4+4)=8(4+4) = 8. The expression is now 8+42−5imes28+4^2-5 imes 2.
  2. Exponents: 42=164^2 = 16. The expression becomes 8+16−5imes28+16-5 imes 2.
  3. Multiplication: 5imes2=105 imes 2 = 10. The expression is 8+16−108+16-10.
  4. Addition and Subtraction: 8+16=248+16 = 24. Then 24−10=1424-10 = 14.

So, adding parentheses around 4+44+4 changes the value to 14. This doesn't help us reach 10. In fact, it moves us further away!

Option C: Add parentheses around 42−54^2-5.

Let's see what happens when we group 42−54^2-5: 4+(42−5)−5imes24+(4^2-5)-5 imes 2. Oops, wait, the original expression is 4+42−5imes24+4^2-5 imes 2. So, adding parentheses around 42−54^2-5 would look like 4+(42−5)imes24+(4^2-5) imes 2. My apologies for the slip-up there, guys. Always double-check where those parentheses are going!

Let's try that again: 4+(42−5)imes24+(4^2-5) imes 2.

  1. Parentheses: Inside the parentheses, we have 42−54^2-5. First, the exponent: 42=164^2 = 16. So, inside the parentheses, we have 16−5=1116-5 = 11. The expression is now 4+11imes24+11 imes 2.
  2. Multiplication: Next, 11imes2=2211 imes 2 = 22. The expression becomes 4+224+22.
  3. Addition: 4+22=264+22 = 26.

Adding parentheses around 42−54^2-5 results in 26. This is definitely not 10.

Option D: Add parentheses around 4+42−54+4^2-5.

Finally, let's add parentheses around the first three terms: (4+42−5)imes2(4+4^2-5) imes 2. Let's break it down:

  1. Parentheses: Inside the parentheses, we follow PEMDAS again. First, the exponent: 42=164^2 = 16. So, inside the parentheses, we have 4+16−54+16-5. Now, addition and subtraction from left to right: 4+16=204+16 = 20. Then 20−5=1520-5 = 15. The expression is now (15)imes2(15) imes 2.
  2. Multiplication: 15imes2=3015 imes 2 = 30.

This option gives us 30. Not 10 either.

The Conclusion: Why Option A is the Winner

After carefully evaluating the original expression and each of the proposed modifications, we found the following:

  • Original expression (4+42−5imes24+4^2-5 imes 2): 10
  • Option B ((4+4)+42−5imes2(4+4)+4^2-5 imes 2): 14
  • Option C (4+(42−5)imes24+(4^2-5) imes 2): 26
  • Option D ((4+42−5)imes2(4+4^2-5) imes 2): 30

The question asks what should be done so that the expression will have a value of 10. Since the original expression already has a value of 10 without any modifications, the correct answer is that nothing needs to be done. This highlights how crucial it is to correctly apply the order of operations. Sometimes, the simplest answer is the right one, and you don't need to overcomplicate things by adding unnecessary steps or changes.

So, the next time you're faced with a math problem like this, always start by evaluating it as is. You might be surprised to find you've already hit the target! Keep practicing, keep exploring, and remember the power of PEMDAS!