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Simplify the left-hand side of the equation sqrt(3)+sqrt(5)sqrt(5).

Mathematics

ScanSolve AI Answer

Answer

C

Step-by-step solution

5 steps

To find the value of (x+y)(x+y), we first need to simplify the given expression and then compare it to x+y15x + y\sqrt{15} to find xx and yy.

  1. Simplify the left-hand side of the equation 3+55\frac{\sqrt{3}+\sqrt{5}}{\sqrt{5}}.

    We can split the fraction into two terms:

    3+55=35+55\frac{\sqrt{3}+\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3}}{\sqrt{5}} + \frac{\sqrt{5}}{\sqrt{5}} =35+1= \frac{\sqrt{3}}{\sqrt{5}} + 1
  2. Rationalize the denominator of the term 35\frac{\sqrt{3}}{\sqrt{5}}.

    Multiply the numerator and the denominator by 5\sqrt{5}:

    35×55=3×55=155\frac{\sqrt{3}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3 \times 5}}{5} = \frac{\sqrt{15}}{5}
  3. Substitute the rationalized term back into the expression.

    155+1\frac{\sqrt{15}}{5} + 1

    Rearrange the terms to match the form x+y15x + y\sqrt{15}:

    1+15151 + \frac{1}{5}\sqrt{15}
  4. Compare this simplified expression with x+y15x + y\sqrt{15}.

    1+1515=x+y151 + \frac{1}{5}\sqrt{15} = x + y\sqrt{15}

    By comparing the constant terms and the coefficients of 15\sqrt{15}, we can identify xx and yy:

    x=1x = 1 y=15y = \frac{1}{5}
  5. Calculate the value of (x+y)(x+y).

    x+y=1+15x+y = 1 + \frac{1}{5}

    To add these, find a common denominator:

    x+y=55+15=5+15=65x+y = \frac{5}{5} + \frac{1}{5} = \frac{5+1}{5} = \frac{6}{5}

    Convert the improper fraction to a mixed number:

    65=115\frac{6}{5} = 1\frac{1}{5}

    Comparing this result with the given options:
    A. 1351\frac{3}{5}
    B. 1251\frac{2}{5}
    C. 1151\frac{1}{5}
    D. 15\frac{1}{5}

    The correct option is C.

    The final answer is C. 115\boxed{\text{C. } 1\frac{1}{5}}.

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