Answered

  1. Rewrite this number as a whole number or decimal:

$3.61 \times 10^4$


  1. Multiply and simplify completely:

$(6 x+8)(6 x-8)$


  1. Multiply and simplify:

$2 x^2\left(3 x^2+3 x+4\right)$


  1. Multiply and simplify:

$6 y(4 y+7)(y-2)$

Verified Answer :

  • For scientific notation \(3.61 \times 10^4\), shift the decimal 4 places right to get \(\boxed{36100}\).
  • Using the difference of squares \((a^2 - b^2)\), \((6x + 8)(6x - 8)\) simplifies to \(\boxed{36x^2 - 64}\).
  • Distributing \(2x^2\) in \(2x^2(3x^2 + 3x + 4)\) yields \(\boxed{6x^4 + 6x^3 + 8x^2}\).
  • Multiplying \(6y(4y + 7)(y - 2)\) gives the final polynomial \(\boxed{24y^3 - 6y^2 - 84y}\).

Explanation

  1. Analysis of the Problems

We need to solve four distinct algebra problems involving powers of ten, binomial products, and polynomial expansion. Each problem requires a different algebraic technique.

  1. Problem 1 asks for the evaluation of a number written in scientific notation.
  2. Problem 2 asks for the product of two binomials in the form of a difference of squares.
  3. Problem 3 requires distributing a monomial into a trinomial.
  4. Problem 4 requires the multiplication of three factors, where two are binomials and one is a monomial.
  1. Evaluation of Question 1

To write \(3.61 \times 10^4\) as a whole number or decimal, we shift the decimal point 4 places to the right since the exponent of 10 is 4.
Starting with 3.61:

  • Shift 1 place: 36.1
  • Shift 2 places: 361
  • Shift 3 places: 3610
  • Shift 4 places: 36100

Thus, \(3.61 \times 10^4 = 36100\).

  1. Simplification of Question 2

The expression \((6x + 8)(6x - 8)\) is in the form \((a + b)(a - b)\), which follows the difference of squares identity \((a + b)(a - b) = a^2 - b^2\).
Here, \(a = 6x\) and \(b = 8\).
Thus, \((6x + 8)(6x - 8) = (6x)^2 - 8^2 = 36x^2 - 64\).

  1. Simplification of Question 3

To simplify \(2x^2(3x^2 + 3x + 4)\), we distribute the term \(2x^2\) into the trinomial:
\(= (2x^2)(3x^2) + (2x^2)(3x) + (2x^2)(4)\)
\(= 6x^4 + 6x^3 + 8x^2\).

  1. Simplification of Question 4

To simplify \(6y(4y + 7)(y - 2)\), we first multiply the two binomials \((4y + 7)(y - 2)\):
\((4y + 7)(y - 2) = 4y(y) + 4y(-2) + 7(y) + 7(-2)\)
\(= 4y^2 - 8y + 7y - 14 = 4y^2 - y - 14\).
Now, multiply the result by \(6y\):
\(6y(4y^2 - y - 14) = 6y(4y^2) - 6y(y) - 6y(14)\)
\(= 24y^3 - 6y^2 - 84y\).

  1. Conclusion

The solutions to the given problems are:

  1. \(36100\)
  2. \(36x^2 - 64\)
  3. \(6x^4 + 6x^3 + 8x^2\)
  4. \(24y^3 - 6y^2 - 84y\)

Examples

These algebraic skills are essential for engineering and physics. For instance, converting scientific units like \(3.61 \times 10^4\) meters to standard notation helps in measurement, while expanding polynomials like \(6x^4 + 6x^3 + 8x^2\) is common in modeling the area or volume of shapes where dimensions change according to a variable \(x\). Mastery of these operations builds the foundation for solving complex equations and analyzing functional relationships in real-world engineering projects.

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