Calculate A + B And A – B: Step-by-Step Solutions

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Hey guys! Today, we're diving into a fun math problem where we need to calculate a + b and a – b given some radical expressions. Don't worry, it might look intimidating at first, but we'll break it down step by step so it's super easy to follow. Grab your pencils, and let's get started!

Part a) Calculating a + b and a – b

Breaking Down a: a = 3√150 (8√50 - 10√48 + 7√108 - 6√98)

First off, let's focus on simplifying a. The expression looks like a jumble of square roots, but we can make it much simpler by breaking down each radical into its prime factors. This means expressing the numbers under the square roots as products of their prime factors. For example, √150 can be simplified because 150 can be factored into 2 × 3 × 5 × 5. Let's go through each term:

  • √150: We can write 150 as 2 × 3 × 5², so √150 = √(2 × 3 × 5²) = 5√(2 × 3) = 5√6.
  • √50: Similarly, 50 = 2 × 5², so √50 = √(2 × 5²) = 5√2.
  • √48: 48 can be written as 2⁴ × 3, so √48 = √(2⁴ × 3) = 2²√3 = 4√3.
  • √108: We have 108 = 2² × 3³, so √108 = √(2² × 3³) = 2 × 3√3 = 6√3.
  • √98: And 98 = 2 × 7², so √98 = √(2 × 7²) = 7√2.

Now we substitute these simplified radicals back into the expression for a:

  • a = 3(5√6) [8(5√2) - 10(4√3) + 7(6√3) - 6(7√2)]
  • a = 15√6 [40√2 - 40√3 + 42√3 - 42√2]

Combine like terms inside the brackets:

  • a = 15√6 [(40√2 - 42√2) + (-40√3 + 42√3)]
  • a = 15√6 [-2√2 + 2√3]

Now, distribute the 15√6:

  • a = 15√6(-2√2) + 15√6(2√3)
  • a = -30√(6×2) + 30√(6×3)
  • a = -30√12 + 30√18

Simplify the radicals further:

  • √12 = √(2² × 3) = 2√3
  • √18 = √(2 × 3²) = 3√2

Substitute these back:

  • a = -30(2√3) + 30(3√2)
  • a = -60√3 + 90√2

So, a is simplified to -60√3 + 90√2.

Simplifying b: b = 4√54 (8√72 - 11√32 + 4√75 - 3√192)

Next, let's simplify b using the same approach. Break down each radical into its prime factors:

  • √54: We can write 54 as 2 × 3³, so √54 = √(2 × 3³) = 3√6.
  • √72: 72 = 2³ × 3², so √72 = √(2³ × 3²) = 2 × 3√2 = 6√2.
  • √32: 32 = 2⁵, so √32 = √(2⁵) = 4√2.
  • √75: 75 = 3 × 5², so √75 = √(3 × 5²) = 5√3.
  • √192: 192 = 2⁶ × 3, so √192 = √(2⁶ × 3) = 2³√3 = 8√3.

Substitute these simplified radicals back into the expression for b:

  • b = 4(3√6) [8(6√2) - 11(4√2) + 4(5√3) - 3(8√3)]
  • b = 12√6 [48√2 - 44√2 + 20√3 - 24√3]

Combine like terms inside the brackets:

  • b = 12√6 [(48√2 - 44√2) + (20√3 - 24√3)]
  • b = 12√6 [4√2 - 4√3]

Distribute the 12√6:

  • b = 12√6(4√2) - 12√6(4√3)
  • b = 48√(6×2) - 48√(6×3)
  • b = 48√12 - 48√18

Simplify the radicals further (we already found these above):

  • √12 = 2√3
  • √18 = 3√2

Substitute these back:

  • b = 48(2√3) - 48(3√2)
  • b = 96√3 - 144√2

So, b simplifies to 96√3 - 144√2.

Calculating a + b

Now that we have simplified a and b, let's calculate a + b:

  • a = -60√3 + 90√2
  • b = 96√3 - 144√2
  • a + b = (-60√3 + 90√2) + (96√3 - 144√2)

Combine like terms:

  • a + b = (-60√3 + 96√3) + (90√2 - 144√2)
  • a + b = 36√3 - 54√2

So, a + b = 36√3 - 54√2.

Calculating a – b

Now, let's calculate a – b:

  • a = -60√3 + 90√2
  • b = 96√3 - 144√2
  • a - b = (-60√3 + 90√2) - (96√3 - 144√2)

Distribute the negative sign and combine like terms:

  • a - b = -60√3 + 90√2 - 96√3 + 144√2
  • a - b = (-60√3 - 96√3) + (90√2 + 144√2)
  • a - b = -156√3 + 234√2

So, a – b = -156√3 + 234√2.

Part b) Calculating a + b and a – b

Simplifying a: a = 5√24 (5√75 - 3√243 - 3√432 + 4√192)

Let's simplify a again, this time with a new set of radicals. Breaking down each radical into its prime factors:

  • √24: We can write 24 as 2³ × 3, so √24 = √(2³ × 3) = 2√6.
  • √75: We already know 75 = 3 × 5², so √75 = 5√3.
  • √243: 243 = 3⁵, so √243 = √(3⁵) = 9√3.
  • √432: 432 = 2⁴ × 3³, so √432 = √(2⁴ × 3³) = 12√3.
  • √192: We already know 192 = 2⁶ × 3, so √192 = 8√3.

Substitute these simplified radicals back into the expression for a:

  • a = 5(2√6) [5(5√3) - 3(9√3) - 3(12√3) + 4(8√3)]
  • a = 10√6 [25√3 - 27√3 - 36√3 + 32√3]

Combine like terms inside the brackets:

  • a = 10√6 [(25 - 27 - 36 + 32)√3]
  • a = 10√6 [-6√3]

Now, multiply:

  • a = -60√(6×3)
  • a = -60√18

Simplify the radical further:

  • √18 = 3√2

Substitute this back:

  • a = -60(3√2)
  • a = -180√2

So, a simplifies to -180√2.

The value of b = 3

In this case, b is simply given as 3. There's no radical to simplify!

Calculating a + b

Now that we have simplified a and know b, let's calculate a + b:

  • a = -180√2
  • b = 3
  • a + b = -180√2 + 3

So, a + b = -180√2 + 3.

Calculating a – b

Finally, let's calculate a – b:

  • a = -180√2
  • b = 3
  • a - b = -180√2 - 3

So, a – b = -180√2 - 3.

Final Thoughts

Alright, we made it through both parts of this radical calculation! Remember, the key to tackling these kinds of problems is to break them down into smaller, manageable steps. Simplifying radicals by finding prime factors is super helpful. And don't forget to double-check your work as you go. You guys did awesome, and I hope this helped clear things up. Keep practicing, and you'll become a math whiz in no time!