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Simplify. Remove all perfect squares from inside the square root. Assume y is positive. sqrt(200y^(4))=◻

Simplify.\newlineRemove all perfect squares from inside the square root. Assume y y is positive.\newline200y4=□ \sqrt{200 y^{4}}=\square

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Full solution

Q. Simplify.\newlineRemove all perfect squares from inside the square root. Assume y y is positive.\newline200y4=□ \sqrt{200 y^{4}}=\square
  1. Factorization and Pairing: Factor 200y4200y^4 into its prime factors and pair perfect squares.\newline200y4=2×2×2×5×5×y2×y2200y^4 = 2 \times 2 \times 2 \times 5 \times 5 \times y^2 \times y^2\newlineWe can pair the perfect squares as (2×2)×(5×5)×(y2×y2)(2 \times 2) \times (5 \times 5) \times (y^2 \times y^2).
  2. Rewriting the Square Root: Rewrite the square root of the product of perfect squares and the remaining factor. 200y4=(2×2)×(5×5)×(y2×y2)\sqrt{200y^4} = \sqrt{(2 \times 2) \times (5 \times 5) \times (y^2 \times y^2)}
  3. Taking the Square Root: Take the square root of the perfect squares. (2×2)×(5×5)×(y2×y2)=22×52×y2×y2\sqrt{(2 \times 2) \times (5 \times 5) \times (y^2 \times y^2)} = \sqrt{2^2} \times \sqrt{5^2} \times \sqrt{y^2 \times y^2}=2×5×y×y= 2 \times 5 \times y \times y=10y2= 10y^2

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