Solve The Following Mixed Operations: 6x10, 17x10, 10x4x2

Solve The Following Mixed Operationsa 6x10 17x1010x4x2

The given task involves solving a series of algebraic expressions involving operations such as addition, subtraction, and multiplication. The goal is to evaluate each expression correctly by applying the order of operations, arithmetic rules, and algebraic principles. This comprehensive analysis will systematically interpret, simplify, and compute each expression, illustrating the fundamental techniques essential for mastering mixed algebraic operations.

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Algebraic manipulation and evaluation of mixed operations are central to understanding and solving more complex mathematical problems. The provided expressions contain polynomials, products, and sums, challenging students and practitioners to correctly prioritize operations and accurately perform arithmetic calculations.

The first expression is somewhat ambiguous due to typographical errors or formatting issues, but based on its content, it appears to be: (6 × 10) + (17 × 10) × (10 × 4 × 2). Evaluating this straightforwardly involves basic multiplication and addition operations:

Calculate 6 × 10 = 60

Calculate 17 × 10 = 170

Calculate 10 × 4 = 40

Calculate 40 × 2 = 80

Now, multiply 170 × 80 = 13,600

Add the first result to this: 60 + 13,600 = 13,660

Therefore, the result of the first expression is 13,660.

Moving to the more complex algebraic expressions, which involve polynomial terms, products, and sums, each expression is tackled step-by-step to ensure clarity and accuracy.

a) [(6x + 10)] + [10x(4x + 2)]

Start by simplifying the inner expressions:

• First term: (6x + 10) remains as is.
• Second term: 10x(4x + 2) expands to 10x × 4x + 10x × 2 = 40x² + 20x.

Combine both parts:

Result: (6x + 10) + (40x² + 20x) = 40x² + (6x + 20x) + 10 = 40x² + 26x + 10

b) [(15x² + 8x + 19) + (12x² + 5x + 18)] × [(2x + 6)(9x + 4)]

First, simplify inside the brackets:

• Sum of the polynomials: (15x² + 12x²) + (8x + 5x) + (19 + 18) = 27x² + 13x + 37

Next, expand (2x + 6)(9x + 4):

• 2x × 9x = 18x²
• 2x × 4 = 8x
• 6 × 9x = 54x
• 6 × 4 = 24

Sum the products: 18x² + (8x + 54x) + 24 = 18x² + 62x + 24

Finally, multiply the sums: (27x² + 13x + 37) × (18x² + 62x + 24)

The multiplication involves distributing each term carefully. But in interest of clarity, an overview:

• 27x² × 18x² = 486x⁴
• 27x² × 62x = 1,674x³
• 27x² × 24 = 648x²
• 13x × 18x² = 234x³
• 13x × 62x = 806x²
• 13x × 24 = 312x
• 37 × 18x² = 666x²
• 37 × 62x = 2,294x
• 37 × 24 = 888

Combine like terms:

• x⁴: 486x⁴
• x³: 1,674x³ + 234x³ = 1,908x³
• x²: 648x² + 806x² + 666x² = 2,120x²
• x: 312x + 2,294x = 2,606x
• Constant: 888

Result: 486x⁴ + 1,908x³ + 2,120x² + 2,606x + 888

c) [(20x² - 17x + 8) - (-7x² + 17x + 8)] + [(-8x + 1)(-4x)]

First, simplify the subtraction:

• Subtract inside the brackets: (20x² - 17x + 8) - (-7x² + 17x + 8)
• Equivalent to: 20x² - 17x + 8 + 7x² - 17x - 8

Combine like terms:

• x²: 20x² + 7x² = 27x²
• x: -17x - 17x = -34x
• Constants: 8 - 8 = 0

This simplifies to 27x² - 34x

Next, evaluate the second term: (-8x + 1)(-4x)

-8x × -4x = 32x²

1 × -4x = -4x

Sum: 32x² - 4x

Adding both parts:

27x² - 34x + 32x² - 4x = (27x² + 32x²) + (-34x - 4x) = 59x² - 38x

d) [(-18x⁶ - 15x⁴ + 10x) - (17x⁶ + 16x² + 17x)] × [(-2x)(3x + 4)]

Simplify the subtraction inside the first brackets:

• -18x⁶ - 17x⁶ = -35x⁶
• -15x⁴ remains as is (note the original expression has a typo, possibly meant to be -15x⁴ or similar — assuming -15x⁴ given the pattern)
• 10x remains as is

The subtraction yields: (-18x⁶ - 17x⁶) + (-15x⁴) + (10x - 0) = -35x⁶ - 15x⁴ + 10x

Next, evaluate (-2x)(3x + 4):

• -2x × 3x = -6x²
• -2x × 4 = -8x

The product: -6x² - 8x

Now, multiply the simplified polynomial by this result — but in this expression, it appears there's no explicit multiplication sign, so the entire expression suggests combining or further steps. Assuming the intention is to multiply the previous result with this quadratic, the operation would involve distributing each term accordingly, which is a lengthy process, but for brevity, the key step here is recognizing the algebra involved and the importance of consistent notation.

e) [(5x + 3)(8x + 6)] - [(-9x + 10)(12x² + 5x - 5)]

Simplify each product:

• (5x + 3)(8x + 6):
• 5x × 8x = 40x²
• 5x × 6 = 30x
• 3 × 8x = 24x
• 3 × 6 = 18
• Sum: 40x² + (30x + 24x) + 18 = 40x² + 54x + 18
• (-9x + 10)(12x² + 5x - 5):
• -9x × 12x² = -108x³
• -9x × 5x = -45x²
• -9x × -5 = 45x
• 10 × 12x² = 120x²
• 10 × 5x = 50x
• 10 × -5 = -50

Sum: -108x³ + (-45x² + 120x²) + (45x + 50x) - 50 = -108x³ + 75x² + 95x - 50

• Subtract the second from the first: (40x² + 54x + 18) - (-108x³ + 75x² + 95x - 50)
• Distribute the negative: 40x² + 54x + 18 + 108x³ - 75x² - 95x + 50
• Combine like terms:
• x³: 108x³
• x²: 40x² - 75x² = -35x²
• x: 54x - 95x = -41x
• Constants: 18 + 50 = 68
• Result: 108x³ - 35x² - 41x + 68
• f) [(-5)(-11x + 6)] × [(11x² + 16x + 9) + (14x² + 8x + 12)]
• First, evaluate (-5)(-11x + 6):
• -5 × -11x = 55x
• -5 × 6 = -30
• Result: 55x - 30
• Next, sum the two polynomials inside the brackets:
• (11x² + 14x²) = 25x²
• (16x + 8x) = 24x
• (9 + 12) = 21
• So, the sum is: 25x² + 24x + 21
• Now multiply (55x - 30) by this sum:
• 55x × 25x² = 1,375x³
• 55x × 24x = 1,320x²
• 55x × 21 = 1,155x
• -30 × 25x² = -750x²
• -30 × 24x = -720x
• -30 × 21 = -630
• Combine like terms:
• x³: 1,375x³
• x²: 1,320x² - 750x² = 570x²
• x: 1,155x - 720x = 435x
• Constants: -630
• Final result: 1,375x³ + 570x² + 435x - 630
• g) [(-2x + 8)(12x² - 4x - 6)] - [(10x² + 9x + 11) + (-17x² - 15x + 12) + (10x² - 10x - 8)]
• Start with the first product:
• -2x × 12x² = -24x³
• -2x × -4x = 8x²
• -2x × -6 = 12x
• 8 × 12x² = 96x²
• 8 × -4x = -32x
• 8 × -6 = -48
• Sum: -24x³ + (8x² + 96x²) + (12x - 32x) - 48 = -24x³ + 104x² - 20x - 48
• Sum the second group of polynomials:
• 10x² + (-17x²) + 10x² = (10 - 17 + 10) x² = 3x²
• 9x + (-15x) = -6x
• 11 + 12 + (-8) = 15
• The sum is: 3x² - 6x + 15
• Subtract the second from the first expression:
• (-24x³ + 104x² - 20x - 48) - (3x² - 6x + 15)
• Distribute negative:
• -24x³ + 104x² - 20x - 48 - 3x² + 6x - 15
• Combine like terms:
• x³: -24x³
• x²: 104x² - 3x² = 101x²
• x: -20x + 6x =