Math  /  Algebra

Question(B) Simplify 5326+52\frac{5-3 \sqrt{2}}{6+5 \sqrt{2}} (2. Express 25+522552\frac{2 \sqrt{5}+5 \sqrt{2}}{2 \sqrt{5}-5 \sqrt{2}} in the form (9i) q+r5q+r \sqrt{5}, where q,rq, r and ss are rational number

Studdy Solution

STEP 1

1. Both expressions involve simplifying fractions with radicals in the numerator and denominator.
2. The simplification process will require rationalizing the denominator.
3. The final expressions should be in the simplified form as specified.

STEP 2

1. Simplify 5326+52\frac{5-3 \sqrt{2}}{6+5 \sqrt{2}}.
2. Express 25+522552\frac{2 \sqrt{5}+5 \sqrt{2}}{2 \sqrt{5}-5 \sqrt{2}} in the form q+r5q + r \sqrt{5}, where qq and rr are rational numbers.

STEP 3

To simplify 5326+52\frac{5-3 \sqrt{2}}{6+5 \sqrt{2}}, multiply the numerator and the denominator by the conjugate of the denominator.
5326+52652652 \frac{5-3 \sqrt{2}}{6+5 \sqrt{2}} \cdot \frac{6-5 \sqrt{2}}{6-5 \sqrt{2}}

STEP 4

Expand the numerator using the distributive property:
(532)(652)=56+5(52)+(32)6+(32)(52) (5-3 \sqrt{2})(6-5 \sqrt{2}) = 5 \cdot 6 + 5 \cdot (-5 \sqrt{2}) + (-3 \sqrt{2}) \cdot 6 + (-3 \sqrt{2}) \cdot (-5 \sqrt{2})

STEP 5

Calculate each term in the expansion:
=30252182+152 = 30 - 25 \sqrt{2} - 18 \sqrt{2} + 15 \cdot 2 =30432+30 = 30 - 43 \sqrt{2} + 30 =60432 = 60 - 43 \sqrt{2}

STEP 6

Expand the denominator using the difference of squares:
(6+52)(652)=62(52)2 (6+5 \sqrt{2})(6-5 \sqrt{2}) = 6^2 - (5 \sqrt{2})^2 =36252 = 36 - 25 \cdot 2 =3650 = 36 - 50 =14 = -14

STEP 7

Combine the simplified numerator and denominator:
6043214 \frac{60 - 43 \sqrt{2}}{-14} =601443214 = \frac{60}{-14} - \frac{43 \sqrt{2}}{-14} =307+43214 = -\frac{30}{7} + \frac{43 \sqrt{2}}{14}

STEP 8

Final simplified expression:
5326+52=307+43214 \frac{5-3 \sqrt{2}}{6+5 \sqrt{2}} = -\frac{30}{7} + \frac{43 \sqrt{2}}{14}

STEP 9

To express 25+522552\frac{2 \sqrt{5} + 5 \sqrt{2}}{2 \sqrt{5} - 5 \sqrt{2}} in the form q+r5q + r \sqrt{5}, multiply the numerator and the denominator by the conjugate of the denominator.
25+52255225+5225+52 \frac{2 \sqrt{5} + 5 \sqrt{2}}{2 \sqrt{5} - 5 \sqrt{2}} \cdot \frac{2 \sqrt{5} + 5 \sqrt{2}}{2 \sqrt{5} + 5 \sqrt{2}}

STEP 10

Expand the numerator using the distributive property:
(25+52)(25+52)=(25)2+2552+5225+(52)2 (2 \sqrt{5} + 5 \sqrt{2})(2 \sqrt{5} + 5 \sqrt{2}) = (2 \sqrt{5})^2 + 2 \sqrt{5} \cdot 5 \sqrt{2} + 5 \sqrt{2} \cdot 2 \sqrt{5} + (5 \sqrt{2})^2 =45+1010+1010+252 = 4 \cdot 5 + 10 \sqrt{10} + 10 \sqrt{10} + 25 \cdot 2 =20+2010+50 = 20 + 20 \sqrt{10} + 50 =70+2010 = 70 + 20 \sqrt{10}

STEP 11

Expand the denominator using the difference of squares:
(25)2(52)2 (2 \sqrt{5})^2 - (5 \sqrt{2})^2 =45252 = 4 \cdot 5 - 25 \cdot 2 =2050 = 20 - 50 =30 = -30

STEP 12

Combine the simplified numerator and denominator:
70+201030 \frac{70 + 20 \sqrt{10}}{-30} =7030+201030 = \frac{70}{-30} + \frac{20 \sqrt{10}}{-30} =732103 = -\frac{7}{3} - \frac{2 \sqrt{10}}{3}

STEP 13

Final simplified expression in the form q+r5q + r \sqrt{5}:
25+522552=732103 \frac{2 \sqrt{5} + 5 \sqrt{2}}{2 \sqrt{5} - 5 \sqrt{2}} = -\frac{7}{3} - \frac{2 \sqrt{10}}{3}

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