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Given sqrt(3)+sqrt(5)sqrt(5) = (x+y)sqrt(15), find (x+y).

Mathematics

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2 parts
i

15+1515\frac{1}{5} + \frac{\sqrt{15}}{15}

ii
0

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7 steps

Here are the solutions to the problems.

Part i· 5 steps

Given 3+55=(x+y)15\frac{\sqrt{3}+\sqrt{5}}{\sqrt{5}} = (x+y)\sqrt{15}, find (x+y)(x+y).

  1. Simplify the left-hand side (LHS) of the equation.

    3+55=35+55=35+1\frac{\sqrt{3}+\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3}}{\sqrt{5}} + \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3}}{\sqrt{5}} + 1

  2. Rationalize the denominator of the first term in the LHS.

    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}

    So, the LHS becomes:

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

  3. Equate the simplified LHS to the right-hand side (RHS) and solve for (x+y)(x+y).

    155+1=(x+y)15\frac{\sqrt{15}}{5} + 1 = (x+y)\sqrt{15}

    To isolate (x+y)(x+y), divide both sides by 15\sqrt{15}:

    (x+y)=155+115(x+y) = \frac{\frac{\sqrt{15}}{5} + 1}{\sqrt{15}}

    Combine the terms in the numerator:

    (x+y)=15+5515=15+5515(x+y) = \frac{\frac{\sqrt{15} + 5}{5}}{\sqrt{15}} = \frac{\sqrt{15} + 5}{5\sqrt{15}}

  4. Rationalize the denominator for the expression of (x+y)(x+y).

    Multiply the numerator and denominator by 15\sqrt{15}:

    (x+y)=15+5515×1515(x+y) = \frac{\sqrt{15} + 5}{5\sqrt{15}} \times \frac{\sqrt{15}}{\sqrt{15}} (x+y)=(15)(15)+5155(15)(15)(x+y) = \frac{(\sqrt{15})(\sqrt{15}) + 5\sqrt{15}}{5(\sqrt{15})(\sqrt{15})} (x+y)=15+5155×15(x+y) = \frac{15 + 5\sqrt{15}}{5 \times 15} (x+y)=15+51575(x+y) = \frac{15 + 5\sqrt{15}}{75}

  5. Simplify the expression by dividing each term in the numerator by the denominator.

    (x+y)=1575+51575(x+y) = \frac{15}{75} + \frac{5\sqrt{15}}{75} (x+y)=15+1515(x+y) = \frac{1}{5} + \frac{\sqrt{15}}{15}

    The value of (x+y)(x+y) is:

    15+1515\boxed{\frac{1}{5} + \frac{\sqrt{15}}{15}}

Part ii0

Find 15+55−15+55\frac{\sqrt{15}+5}{5} - \frac{\sqrt{15}+5}{5}

  1. Identify the terms in the expression.

    The expression consists of two identical terms being subtracted from each other.
    Let A=15+55A = \frac{\sqrt{15}+5}{5}. The expression is A−AA - A.

  2. Perform the subtraction.

    When an expression is subtracted from itself, the result is zero.

    15+55−15+55=0\frac{\sqrt{15}+5}{5} - \frac{\sqrt{15}+5}{5} = 0

    The simplified expression is:

    0\boxed{0}
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