Question

simplify the following expression completely, where ( x geq 0 ).
( xsqrt{5xy^4} + sqrt{405x^3y^4} - sqrt{80x^3y^4} )

( -3xy^2sqrt{5x} + xy^2sqrt{405x} )

( 6xy^2sqrt{5x} )

( xy^2sqrt{5x} + xy^2sqrt{405x} - xy^2sqrt{80x} )

( 6xysqrt{5xy^2} )

Explanation:

Step1: Simplify each radical term

• For \( x\sqrt{5xy^4} \): Since \( y^4=(y^2)^2 \) and \( x\geq0 \), we have \( x\sqrt{5xy^4}=x\cdot y^2\sqrt{5x}=xy^2\sqrt{5x} \).
• For \( \sqrt{405x^3y^4} \): Factor \( 405 = 81\times5 \), \( x^3=x^2\cdot x \), \( y^4=(y^2)^2 \). So \( \sqrt{405x^3y^4}=\sqrt{81\times5\times x^2\cdot x\times(y^2)^2}=9xy^2\sqrt{5x} \).
• For \( \sqrt{80x^3y^4} \): Factor \( 80 = 16\times5 \), \( x^3=x^2\cdot x \), \( y^4=(y^2)^2 \). So \( \sqrt{80x^3y^4}=\sqrt{16\times5\times x^2\cdot x\times(y^2)^2}=4xy^2\sqrt{5x} \).

Step2: Substitute back into the original expression

The original expression is \( x\sqrt{5xy^4}+\sqrt{405x^3y^4}-\sqrt{80x^3y^4} \). Substituting the simplified terms, we get:
\( xy^2\sqrt{5x}+9xy^2\sqrt{5x}-4xy^2\sqrt{5x} \)

Step3: Combine like terms

Combine the coefficients of \( xy^2\sqrt{5x} \): \( (1 + 9-4)xy^2\sqrt{5x}=6xy^2\sqrt{5x} \)

Answer:

\( 6xy^2\sqrt{5x} \) (corresponding to the second option in the given choices)