If 1/(3-"√"5) - (2+"√"5)/(3+"√"5) = a + b"√"c, find the value of a, b and c.

Mathematics
If 1/(3-"√"5) - (2+"√"5)/(3+"√"5) = a + b"√"c, find the value of a, b and c.

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Answer

74\frac{7}{4}

Step 1: Rationalize the first term 13−5\frac{1}{3-\sqrt{5}}. To rationalize the denominator, multiply the numerator and denominator by the conjugate of 3−53-\sqrt{5}, which is 3+53+\sqrt{5}. 13−5=13−5×3+53+5\frac{1}{3-\sqrt{5}} = \frac{1}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}} =3+5(3)2−(5)2= \frac{3+\sqrt{5}}{(3)^2 - (\sqrt{5})^2} =3+59−5= \frac{3+\sqrt{5}}{9 - 5} =3+54= \frac{3+\sqrt{5}}{4}

Step 2: Rationalize the second term 2+253+5\frac{2+2\sqrt{5}}{3+\sqrt{5}}. To rationalize the denominator, multiply the numerator and denominator by the conjugate of 3+53+\sqrt{5}, which is 3−53-\sqrt{5}. 2+253+5=2+253+5×3−53−5\frac{2+2\sqrt{5}}{3+\sqrt{5}} = \frac{2+2\sqrt{5}}{3+\sqrt{5}} \times \frac{3-\sqrt{5}}{3-\sqrt{5}} =(2+25)(3−5)(3)2−(5)2= \frac{(2+2\sqrt{5})(3-\sqrt{5})}{(3)^2 - (\sqrt{5})^2} Expand the numerator: (2+25)(3−5)=2(3)−2(5)+25(3)−25(5)(2+2\sqrt{5})(3-\sqrt{5}) = 2(3) - 2(\sqrt{5}) + 2\sqrt{5}(3) - 2\sqrt{5}(\sqrt{5}) =6−25+65−2(5)= 6 - 2\sqrt{5} + 6\sqrt{5} - 2(5) =6+45−10= 6 + 4\sqrt{5} - 10 =−4+45= -4 + 4\sqrt{5} The denominator is 9−5=49-5=4. So, the second term becomes: −4+454=4(−1+5)4\frac{-4+4\sqrt{5}}{4} = \frac{4(-1+\sqrt{5})}{4} =−1+5= -1+\sqrt{5}

Step 3: Substitute the simplified terms back into the original equation. 3+54−(−1+5)=a+bc\frac{3+\sqrt{5}}{4} - (-1+\sqrt{5}) = a+b\sqrt{c} 3+54+1−5=a+bc\frac{3+\sqrt{5}}{4} + 1 - \sqrt{5} = a+b\sqrt{c}

Step 4: Combine the terms on the left-hand side. Find a common denominator, which is 4. 3+54+44−454=a+bc\frac{3+\sqrt{5}}{4} + \frac{4}{4} - \frac{4\sqrt{5}}{4} = a+b\sqrt{c} 3+5+4−454=a+bc\frac{3+\sqrt{5} + 4 - 4\sqrt{5}}{4} = a+b\sqrt{c} Combine the rational parts and the irrational parts: (3+4)+(5−45)4=a+bc\frac{(3+4) + (\sqrt{5} - 4\sqrt{5})}{4} = a+b\sqrt{c} 7−354=a+bc\frac{7 - 3\sqrt{5}}{4} = a+b\sqrt{c}

Step 5: Separate the rational and irrational parts to match the form a+bca+b\sqrt{c}. 74−354=a+bc\frac{7}{4} - \frac{3\sqrt{5}}{4} = a+b\sqrt{c} This can be written as: 74+(−34)5=a+bc\frac{7}{4} + \left(-\frac{3}{4}\right)\sqrt{5} = a+b\sqrt{c}

Step 6: Compare the terms to find the values of aa, bb, and cc. By comparing the rational parts: a=74a = \frac{7}{4} By comparing the coefficients of 5\sqrt{5}: b=−34b = -\frac{3}{4} By comparing the radicands: c=5c = 5

The values are: a = \frac{7{4}} b = -\frac{3{4}} c=5c = \boxed{5}

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Rationalize the first term (1)/(3-sqrt(5)). To rationalize the denominator, multiply the numerator and denominator by the conjugate of 3-sqrt(5), which is 3+sqrt(5).

If 1/(3-"√"5) - (2+"√"5)/(3+"√"5) = a + b"√"c, find the value of a, b and c.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

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Step 1: Rationalize the first term (1)/(3-sqrt(5)). To rationalize the denominator, multiply the numerator and denominator by the conjugate of 3-sqrt(5), which is 3+sqrt(5). (1)/(3-sqrt(5)) = (1)/(3-sqrt(5)) × 3+sqrt(5)3+sqrt(5) = 3+sqrt(5)(3)^2 - (sqrt(5))^2 = 3+sqrt(5)9 - 5 = 3+sqrt(5)4 Step 2: Rationalize the second term 2+2sqrt(5)3+sqrt(5). To rationalize the denominator, multiply the numerator and denominator by the conjugate of 3+sqrt(5), which is 3-sqrt(5). 2+2sqrt(5)3+sqrt(5) = 2+2sqrt(5)3+sqrt(5) × 3-sqrt(5)3-sqrt(5) = (2+2sqrt(5))(3-sqrt(5))(3)^2 - (sqrt(5))^2 Expand the numerator: (2+2sqrt(5))(3-sqrt(5)) = 2(3) - 2(sqrt(5)) + 2sqrt(5)(3) - 2sqrt(5)(sqrt(5)) = 6 - 2sqrt(5) + 6sqrt(5) - 2(5) = 6 + 4sqrt(5) - 10 = -4 + 4sqrt(5) The denominator is 9-5=4. So, the second term becomes: -4+4sqrt(5)4 = 4(-1+sqrt(5))4 = -1+sqrt(5) Step 3: Substitute the simplified terms back into the original equation. 3+sqrt(5)4 - (-1+sqrt(5)) = a+bsqrt(c) 3+sqrt(5)4 + 1 - sqrt(5) = a+bsqrt(c) Step 4: Combine the terms on the left-hand side. Find a common denominator, which is 4. 3+sqrt(5)4 + (4)/(4) - 4sqrt(5)4 = a+bsqrt(c) 3+sqrt(5) + 4 - 4sqrt(5)4 = a+bsqrt(c) Combine the rational parts and the irrational parts: (3+4) + (sqrt(5) - 4sqrt(5))4 = a+bsqrt(c) 7 - 3sqrt(5)4 = a+bsqrt(c) Step 5: Separate the rational and irrational parts to match the form a+bsqrt(c). (7)/(4) - 3sqrt(5)4 = a+bsqrt(c) This can be written as: (7)/(4) + (-(3)/(4))sqrt(5) = a+bsqrt(c) Step 6: Compare the terms to find the values of a, b, and c. By comparing the rational parts: a = (7)/(4) By comparing the coefficients of sqrt(5): b = -(3)/(4) By comparing the radicands: c = 5 The values are: a = (7)/(4) b = -(3)/(4) c = 5