Full solution

\sqrt{16 x^{2} y^{16}+25 x^{16} y^{2}}

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x

and

y

are positive, which of the following is equivalent to the given expression?

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Choose

1

answer:

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(A)

4 x y^{8}+5 x^{8} y

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(B)

4 x y^{4}+5 x^{4} y

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(C)

x y \sqrt{16 y^{14}+25 x^{14}}

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(D)

x^{2} y^{2} \sqrt{16 y^{14}+25 x^{14}}

Recognize terms in expression: Evaluate the square root of the algebraic expression: $\sqrt{16x^{2}y^{16}+25x^{16}y^{2}}$ . 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.
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 $16$ and $25$ and the variables are raised to even powers. $\newline$ The square of a binomial $(a+b)^2$ is $a^2 + 2ab + b^2$ .
Factor out common terms: We notice that the given expression lacks the middle term $2ab$ 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.
Factor out $x^2y^2$ : However, we can still factor out the common terms from each part of the expression under the square root. The common terms are $x^2$ and $y^2$ , which are the lowest powers of $x$ and $y$ in the expression.
Take square root of each factor: Factor out $x^2y^2$ from the expression under the square root: $\sqrt{16x^{2}y^{16}+25x^{16}y^{2}} = \sqrt{x^2y^2(16y^{14}+25x^{14})}.$
Square root of $x^2y^2$ : Since we are taking the square root of the product, we can take the square root of each factor separately: $\sqrt{x^2y^2(16y^{14}+25x^{14})} = \sqrt{x^2y^2} \times \sqrt{16y^{14}+25x^{14}}$ .
Final expression comparison: The square root of $x^2y^2$ is $xy$ because both $x$ and $y$ are positive: $\sqrt{x^2y^2} = xy$ .
Correct answer identification: Now we have the expression $xy \sqrt{16y^{14}+25x^{14}}$ . This expression cannot be simplified further because the term inside the square root does not factor into a square of a binomial.
Correct answer identification: Now we have the expression $xy \sqrt{16y^{14}+25x^{14}}$ . 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 $xy \sqrt{16y^{14}+25x^{14}}$ .
Correct answer identification: Now we have the expression $xy \sqrt{16y^{14}+25x^{14}}$ . 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 $xy \sqrt{16y^{14}+25x^{14}}$ .The correct answer is (D) $x^{2}y^{2}\sqrt{16y^{14}+25x^{14}}$ , which simplifies to $xy \sqrt{16y^{14}+25x^{14}}$ when we take the square root of $x^2y^2$ .