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(2+(1)/(3)d)(5d+(1)/(3)d^(2)) Which of the following expressions is equivalent to the given expression? Choose 1 answer: (A) 2+2d^(2) (B) 2+(16)/(3)d+(1)/(3)d^(2) (c) 2+(5)/(3)d^(2)+(1)/(9)d^(3) (D) 10 d+(7)/(3)d^(2)+(1)/(9)d^(3)

(2+13d)(5d+13d2) \left(2+\frac{1}{3} d\right)\left(5 d+\frac{1}{3} d^{2}\right) \newlineWhich of the following expressions is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) 2+2d2 2+2 d^{2} \newline(B) 2+163d+13d2 2+\frac{16}{3} d+\frac{1}{3} d^{2} \newline(C) 2+53d2+19d3 2+\frac{5}{3} d^{2}+\frac{1}{9} d^{3} \newline(D) 10d+73d2+19d3 10 d+\frac{7}{3} d^{2}+\frac{1}{9} d^{3}

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Full solution

Q. (2+13d)(5d+13d2) \left(2+\frac{1}{3} d\right)\left(5 d+\frac{1}{3} d^{2}\right) \newlineWhich of the following expressions is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) 2+2d2 2+2 d^{2} \newline(B) 2+163d+13d2 2+\frac{16}{3} d+\frac{1}{3} d^{2} \newline(C) 2+53d2+19d3 2+\frac{5}{3} d^{2}+\frac{1}{9} d^{3} \newline(D) 10d+73d2+19d3 10 d+\frac{7}{3} d^{2}+\frac{1}{9} d^{3}
  1. Apply Distributive Property: First, we will apply the distributive property (also known as the FOIL method for binomials) to multiply the two expressions.\newline(2+13d)(5d+13d2)=2⋅(5d)+2⋅(13d2)+13d⋅(5d)+13d⋅(13d2)(2+\frac{1}{3}d)(5d+\frac{1}{3}d^{2}) = 2\cdot(5d) + 2\cdot\left(\frac{1}{3}d^{2}\right) + \frac{1}{3}d\cdot(5d) + \frac{1}{3}d\cdot\left(\frac{1}{3}d^{2}\right)
  2. Perform Multiplication: Now, we will perform the multiplication for each term.\newline2×(5d)=10d2\times(5d) = 10d\newline2×(13d2)=23d22\times\left(\frac{1}{3}d^{2}\right) = \frac{2}{3}d^{2}\newline13d×(5d)=53d2\frac{1}{3}d\times(5d) = \frac{5}{3}d^{2}\newline13d×(13d2)=19d3\frac{1}{3}d\times\left(\frac{1}{3}d^{2}\right) = \frac{1}{9}d^{3}
  3. Combine Like Terms: Next, we will combine like terms. 10d+(23)d2+(53)d2+(19)d310d + \left(\frac{2}{3}\right)d^{2} + \left(\frac{5}{3}\right)d^{2} + \left(\frac{1}{9}\right)d^{3}
  4. Add Coefficients: Adding the coefficients of the d2d^2 terms:\newline23d2+53d2=(23+53)d2=73d2\frac{2}{3}d^{2} + \frac{5}{3}d^{2} = \left(\frac{2}{3} + \frac{5}{3}\right)d^{2} = \frac{7}{3}d^{2}
  5. Simplified Expression: Now, we have the simplified expression: 10d+(73)d2+(19)d310d + \left(\frac{7}{3}\right)d^{2} + \left(\frac{1}{9}\right)d^{3}
  6. Compare with Options: We compare the simplified expression with the given options to find the match.\newlineThe correct option is (D) 10d+73d2+19d310d + \frac{7}{3}d^{2} + \frac{1}{9}d^{3}.

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