Verify Equality: M^6 + N^6 = 2m^3n^3 + (m^3 - N^3)^2

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Verify Equality: m^6 + n^6 = 2m^3n^3 + (m^3 - n^3)^2

Hey guys! Today, we're diving into a fun little math problem where we need to verify if the equation m^6 + n^6 = 2m3n3 + (m^3 - n3)2 holds true for different values of m and n. We’ll go through each case step by step, so you can follow along easily. Grab your calculators, and let's get started!

a) m = 3 and n = 1

Let's kick things off by plugging in m = 3 and n = 1 into our equation:

Original Equation: m^6 + n^6 = 2m3n3 + (m^3 - n3)2

Substitute the values:

3^6 + 1^6 = 2(33)(13) + (3^3 - 13)2

Now, let's break it down:

  • 3^6 = 729
  • 1^6 = 1
  • 2(33)(13) = 2 * 27 * 1 = 54
  • 3^3 - 1^3 = 27 - 1 = 26
  • (3^3 - 13)2 = 26^2 = 676

Plug these back into the equation:

729 + 1 = 54 + 676

730 = 730

Verdict: The equation holds true for m = 3 and n = 1. Awesome!

Detailed Explanation

Okay, let's really break down why this works. When we substitute m = 3 and n = 1, we're essentially checking if both sides of the equation balance out. On the left side, we have 3^6 which is 729, and 1^6 which is just 1. Adding them gives us 730. On the right side, we first calculate 2 * 3^3 * 1^3, which simplifies to 2 * 27 * 1 = 54. Then, we find the difference between 3^3 and 1^3, which is 27 - 1 = 26. Squaring 26 gives us 676. Adding 54 and 676 also gives us 730. Since both sides are equal, the equation is indeed valid for these values.

The key here is understanding the order of operations (PEMDAS/BODMAS). We deal with exponents first, then multiplication, and finally addition and subtraction. By following this order, we ensure that we're calculating each term correctly. This methodical approach helps prevent errors and makes the verification process much smoother.

Also, note that the equation itself is a specific instance of a more general algebraic identity. Recognizing these patterns can be super helpful in solving similar problems quickly. For instance, if you remember algebraic identities like (a - b)^2 = a^2 - 2ab + b^2, you'll start to see how the given equation is structured and how the terms relate to each other. This kind of pattern recognition can make math problems a lot less intimidating!

b) m = 2 and n = 1

Next up, let's see if the equation holds when m = 2 and n = 1.

Original Equation: m^6 + n^6 = 2m3n3 + (m^3 - n3)2

Substitute the values:

2^6 + 1^6 = 2(23)(13) + (2^3 - 13)2

Time to break it down:

  • 2^6 = 64
  • 1^6 = 1
  • 2(23)(13) = 2 * 8 * 1 = 16
  • 2^3 - 1^3 = 8 - 1 = 7
  • (2^3 - 13)2 = 7^2 = 49

Plug these values back into the equation:

64 + 1 = 16 + 49

65 = 65

Verdict: The equation also holds true for m = 2 and n = 1. Fantastic!

Extended Discussion

Let's dig a bit deeper into why this case also works out perfectly. When we replace m with 2 and n with 1, the equation transforms into something we can easily compute. On the left-hand side, we have 2^6, which equals 64, and 1^6, which is just 1. Together, they sum up to 65.

On the right-hand side, we have 2 * (2^3) * (1^3), which simplifies to 2 * 8 * 1 = 16. Then, we calculate (2^3 - 13)2, which is (8 - 1)^2 = 7^2 = 49. Adding 16 and 49 also gives us 65. The equality 65 = 65 confirms that the equation holds true for these values of m and n.

What's important here is to maintain accuracy in each step. Exponents, multiplication, subtraction, and addition all need to be performed in the correct order to avoid errors. This careful approach ensures that the final result is correct and reliable.

Notice how changing the values of m and n changes the individual terms, but the overall structure of the equation allows it to remain balanced. This is a fundamental aspect of algebraic identities: they hold true for a wide range of values, making them incredibly useful in mathematical problem-solving. Recognizing and understanding these identities can significantly speed up your ability to tackle complex problems.

c) m = 2 and n = 2

Now, let's test the equation with m = 2 and n = 2.

Original Equation: m^6 + n^6 = 2m3n3 + (m^3 - n3)2

Substitute the values:

2^6 + 2^6 = 2(23)(23) + (2^3 - 23)2

Let's break it down:

  • 2^6 = 64
  • 2(23)(23) = 2 * 8 * 8 = 128
  • 2^3 - 2^3 = 8 - 8 = 0
  • (2^3 - 23)2 = 0^2 = 0

Plug these back into the equation:

64 + 64 = 128 + 0

128 = 128

Verdict: The equation also holds true for m = 2 and n = 2. Sweet!

Expanded Explanation

In this scenario, setting m = 2 and n = 2 leads to some interesting simplifications. The left-hand side of the equation becomes 2^6 + 2^6, which is 64 + 64 = 128. On the right-hand side, we have 2 * (2^3) * (2^3), which is 2 * 8 * 8 = 128. The term (2^3 - 23)2 becomes (8 - 8)^2 = 0^2 = 0.

Putting it all together, we get 128 = 128 + 0, which simplifies to 128 = 128. This equality shows that the equation remains valid when m and n are equal. The zero term on the right side makes the equation trivially true, highlighting an important aspect of algebraic identities: they must hold true even in special cases.

The key takeaway here is that when m = n, the term (m^3 - n3)2 will always be zero, simplifying the equation significantly. Recognizing such patterns can help you quickly verify the equation without needing to compute all the terms individually. This demonstrates the power of understanding algebraic structures and how they behave under different conditions.

d) m = 3 and n = 2

Lastly, let's plug in m = 3 and n = 2 and see what happens.

Original Equation: m^6 + n^6 = 2m3n3 + (m^3 - n3)2

Substitute the values:

3^6 + 2^6 = 2(33)(23) + (3^3 - 23)2

Time to break it down:

  • 3^6 = 729
  • 2^6 = 64
  • 2(33)(23) = 2 * 27 * 8 = 432
  • 3^3 - 2^3 = 27 - 8 = 19
  • (3^3 - 23)2 = 19^2 = 361

Plug these values back into the equation:

729 + 64 = 432 + 361

793 = 793

Verdict: The equation holds true for m = 3 and n = 2. Yes!

Deep Dive Explanation

Let’s break down why this final case also holds true. Substituting m = 3 and n = 2 into the equation, we get 3^6 + 2^6 on the left-hand side, which calculates to 729 + 64 = 793. On the right-hand side, we have 2 * (3^3) * (2^3), which simplifies to 2 * 27 * 8 = 432. The term (3^3 - 23)2 becomes (27 - 8)^2 = 19^2 = 361. Adding 432 and 361 gives us 793.

The equality 793 = 793 confirms that the equation is valid for m = 3 and n = 2. This example reinforces the importance of accurate calculation and attention to detail. Each term must be computed correctly to ensure the final equality holds.

What’s interesting here is that even with different values for m and n, the structure of the equation maintains its balance. This is because the equation is derived from a more fundamental algebraic identity. Understanding these underlying principles allows you to verify the equation quickly and confidently, regardless of the specific values of m and n.

In conclusion, we have successfully verified the equality m^6 + n^6 = 2m3n3 + (m^3 - n3)2 for all the given cases. Keep practicing, and you'll become a math whiz in no time! Keep rocking!