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Simplify the radical in the denominator.

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2 parts
i

410−103\frac{4\sqrt{10} - 10}{3}

ii

6+32+23+266 + 3\sqrt{2} + 2\sqrt{3} + 2\sqrt{6}

Step-by-step solution

9 steps

Here are the solutions to the simplification problems.

Part i· 4 steps

Simplify 258+5\frac{2\sqrt{5}}{\sqrt{8} + \sqrt{5}}

  1. Simplify the radical in the denominator.

    8=4×2=22\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}

    The expression becomes:

    2522+5\frac{2\sqrt{5}}{2\sqrt{2} + \sqrt{5}}

  2. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 22−52\sqrt{2} - \sqrt{5}.

    2522+5×22−522−5\frac{2\sqrt{5}}{2\sqrt{2} + \sqrt{5}} \times \frac{2\sqrt{2} - \sqrt{5}}{2\sqrt{2} - \sqrt{5}}

  3. Expand the numerator and the denominator.

    Numerator:

    25(22−5)=(25)(22)−(25)(5)=410−2(5)=410−102\sqrt{5}(2\sqrt{2} - \sqrt{5}) = (2\sqrt{5})(2\sqrt{2}) - (2\sqrt{5})(\sqrt{5}) = 4\sqrt{10} - 2(5) = 4\sqrt{10} - 10

    Denominator (using the difference of squares formula (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2):

    (22+5)(22−5)=(22)2−(5)2=(4×2)−5=8−5=3(2\sqrt{2} + \sqrt{5})(2\sqrt{2} - \sqrt{5}) = (2\sqrt{2})^2 - (\sqrt{5})^2 = (4 \times 2) - 5 = 8 - 5 = 3

  4. Combine the simplified numerator and denominator.

    410−103\frac{4\sqrt{10} - 10}{3}

    The simplified expression is:

    410−103\boxed{\frac{4\sqrt{10} - 10}{3}}

Part ii

6+32+23+266 + 3\sqrt{2} + 2\sqrt{3} + 2\sqrt{6}

Simplify 2(3+32)32−23\frac{2(3+3\sqrt{2})}{3\sqrt{2} - 2\sqrt{3}}

  1. Distribute the 2 in the numerator.

    2(3+32)=6+622(3+3\sqrt{2}) = 6 + 6\sqrt{2}

    The expression becomes:

    6+6232−23\frac{6 + 6\sqrt{2}}{3\sqrt{2} - 2\sqrt{3}}

  2. Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 32+233\sqrt{2} + 2\sqrt{3}.

    6+6232−23×32+2332+23\frac{6 + 6\sqrt{2}}{3\sqrt{2} - 2\sqrt{3}} \times \frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} + 2\sqrt{3}}

  3. Expand the numerator.

    (6+62)(32+23)=6(32)+6(23)+(62)(32)+(62)(23)(6 + 6\sqrt{2})(3\sqrt{2} + 2\sqrt{3}) = 6(3\sqrt{2}) + 6(2\sqrt{3}) + (6\sqrt{2})(3\sqrt{2}) + (6\sqrt{2})(2\sqrt{3}) =182+123+18(2)+126= 18\sqrt{2} + 12\sqrt{3} + 18(2) + 12\sqrt{6} =182+123+36+126= 18\sqrt{2} + 12\sqrt{3} + 36 + 12\sqrt{6}

    Rearranging the terms:

    36+182+123+12636 + 18\sqrt{2} + 12\sqrt{3} + 12\sqrt{6}

  4. Expand the denominator (using the difference of squares formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2).

    (32−23)(32+23)=(32)2−(23)2(3\sqrt{2} - 2\sqrt{3})(3\sqrt{2} + 2\sqrt{3}) = (3\sqrt{2})^2 - (2\sqrt{3})^2 =(9×2)−(4×3)=18−12=6= (9 \times 2) - (4 \times 3) = 18 - 12 = 6

  5. Combine the simplified numerator and denominator, then simplify by dividing each term in the numerator by the denominator.

    36+182+123+1266\frac{36 + 18\sqrt{2} + 12\sqrt{3} + 12\sqrt{6}}{6} =366+1826+1236+1266= \frac{36}{6} + \frac{18\sqrt{2}}{6} + \frac{12\sqrt{3}}{6} + \frac{12\sqrt{6}}{6} =6+32+23+26= 6 + 3\sqrt{2} + 2\sqrt{3} + 2\sqrt{6}

    The simplified expression is:

    6+32+23+26\boxed{6 + 3\sqrt{2} + 2\sqrt{3} + 2\sqrt{6}}
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