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sqrt(16x^(2)y^(16)+25x^(16)y^(2)) If x and y are positive, which of the following is equivalent to the given expression? Choose 1 answer: (A) 4xy^(8)+5x^(8)y (B) 4xy^(4)+5x^(4)y (C) xysqrt(16y^(14)+25x^(14)) (D) x^(2)y^(2)sqrt(16y^(14)+25x^(14))

16x2y16+25x16y2 \sqrt{16 x^{2} y^{16}+25 x^{16} y^{2}} \newlineIf x x and y y are positive, which of the following is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) 4xy8+5x8y 4 x y^{8}+5 x^{8} y \newline(B) 4xy4+5x4y 4 x y^{4}+5 x^{4} y \newline(C) xy16y14+25x14 x y \sqrt{16 y^{14}+25 x^{14}} \newline(D) x2y216y14+25x14 x^{2} y^{2} \sqrt{16 y^{14}+25 x^{14}}

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Q. 16x2y16+25x16y2 \sqrt{16 x^{2} y^{16}+25 x^{16} y^{2}} \newlineIf x x and y y are positive, which of the following is equivalent to the given expression?\newlineChoose 11 answer:\newline(A) 4xy8+5x8y 4 x y^{8}+5 x^{8} y \newline(B) 4xy4+5x4y 4 x y^{4}+5 x^{4} y \newline(C) xy16y14+25x14 x y \sqrt{16 y^{14}+25 x^{14}} \newline(D) x2y216y14+25x14 x^{2} y^{2} \sqrt{16 y^{14}+25 x^{14}}
  1. Recognize terms in expression: Evaluate the square root of the algebraic expression: 16x2y16+25x16y2\sqrt{16x^{2}y^{16}+25x^{16}y^{2}}. We recognize that the expression inside the square root is a sum of two terms, each of which is a product of a constant and two variables raised to a power.
  2. Pattern of binomial square: We look for a pattern that resembles the square of a binomial because the terms inside the square root have coefficients that are perfect squares 1616 and 2525 and the variables are raised to even powers.\newlineThe square of a binomial (a+b)2(a+b)^2 is a2+2ab+b2a^2 + 2ab + b^2.
  3. Factor out common terms: We notice that the given expression lacks the middle term 2ab2ab that would be present if it were the square of a binomial. This means we cannot directly factor the expression as the square of a binomial.
  4. Factor out x2y2x^2y^2: However, we can still factor out the common terms from each part of the expression under the square root. The common terms are x2x^2 and y2y^2, which are the lowest powers of xx and yy in the expression.
  5. Take square root of each factor: Factor out x2y2x^2y^2 from the expression under the square root: 16x2y16+25x16y2=x2y2(16y14+25x14).\sqrt{16x^{2}y^{16}+25x^{16}y^{2}} = \sqrt{x^2y^2(16y^{14}+25x^{14})}.
  6. Square root of x2y2x^2y^2: Since we are taking the square root of the product, we can take the square root of each factor separately: x2y2(16y14+25x14)=x2y2×16y14+25x14\sqrt{x^2y^2(16y^{14}+25x^{14})} = \sqrt{x^2y^2} \times \sqrt{16y^{14}+25x^{14}}.
  7. Final expression comparison: The square root of x2y2x^2y^2 is xyxy because both xx and yy are positive: x2y2=xy\sqrt{x^2y^2} = xy.
  8. Correct answer identification: Now we have the expression xy16y14+25x14xy \sqrt{16y^{14}+25x^{14}}. This expression cannot be simplified further because the term inside the square root does not factor into a square of a binomial.
  9. Correct answer identification: Now we have the expression xy16y14+25x14xy \sqrt{16y^{14}+25x^{14}}. This expression cannot be simplified further because the term inside the square root does not factor into a square of a binomial.We compare the final expression with the answer choices. The correct answer must be equivalent to xy16y14+25x14xy \sqrt{16y^{14}+25x^{14}}.
  10. Correct answer identification: Now we have the expression xy16y14+25x14xy \sqrt{16y^{14}+25x^{14}}. This expression cannot be simplified further because the term inside the square root does not factor into a square of a binomial.We compare the final expression with the answer choices. The correct answer must be equivalent to xy16y14+25x14xy \sqrt{16y^{14}+25x^{14}}.The correct answer is (D) x2y216y14+25x14x^{2}y^{2}\sqrt{16y^{14}+25x^{14}}, which simplifies to xy16y14+25x14xy \sqrt{16y^{14}+25x^{14}} when we take the square root of x2y2x^2y^2.

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