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To solve the equation [tex]\(\frac{2 + 3 \sqrt{5}}{4 + 5 \sqrt{5}} = a + b \sqrt{5}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are rational numbers, we need to rationalize the denominator.
Here's the step-by-step solution:
1. Given Equation:
[tex]\[ \frac{2+3\sqrt{5}}{4+5\sqrt{5}} \][/tex]
2. Rationalize the Denominator:
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(4 + 5\sqrt{5}\)[/tex] is [tex]\(4 - 5\sqrt{5}\)[/tex].
3. Multiply the Numerator and Denominator by the Conjugate:
[tex]\[ \frac{(2 + 3\sqrt{5})(4 - 5\sqrt{5})}{(4 + 5\sqrt{5})(4 - 5\sqrt{5})} \][/tex]
4. Simplify the Denominator:
Use the difference of squares formula: [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex].
[tex]\[ (4 + 5\sqrt{5})(4 - 5\sqrt{5}) = 4^2 - (5\sqrt{5})^2 = 16 - 125 = -109 \][/tex]
5. Expand and Simplify the Numerator:
[tex]\[ (2 + 3\sqrt{5})(4 - 5\sqrt{5}) = 2 \cdot 4 + 2 \cdot (-5\sqrt{5}) + 3\sqrt{5} \cdot 4 + 3\sqrt{5} \cdot (-5\sqrt{5}) \][/tex]
Simplify each term:
[tex]\[ 2 \cdot 4 = 8, \quad 2 \cdot (-5\sqrt{5}) = -10\sqrt{5}, \quad 3\sqrt{5} \cdot 4 = 12\sqrt{5}, \quad 3\sqrt{5} \cdot (-5\sqrt{5}) = -75 \cdot 5 = -75 \][/tex]
Combine these terms:
[tex]\[ 8 - 75 - 10\sqrt{5} + 12\sqrt{5} \][/tex]
Simplify further:
[tex]\[ -67 + 2\sqrt{5} \][/tex]
6. Form the Fraction:
[tex]\[ \frac{-67 + 2\sqrt{5}}{-109} \][/tex]
7. Separate the Rational and Irrational Parts:
[tex]\[ \frac{-67}{-109} + \frac{2\sqrt{5}}{-109} \][/tex]
Simplify the fractions:
[tex]\[ \frac{-67}{-109} = \frac{67}{109} \quad \text{and} \quad \frac{2\sqrt{5}}{-109} = -\frac{2\sqrt{5}}{109} \][/tex]
Thus, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[ a = \frac{67}{109} \approx 0.6146788990825688, \quad b = -\frac{2}{109} \approx -0.01834862385321101 \][/tex]
Here's the step-by-step solution:
1. Given Equation:
[tex]\[ \frac{2+3\sqrt{5}}{4+5\sqrt{5}} \][/tex]
2. Rationalize the Denominator:
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(4 + 5\sqrt{5}\)[/tex] is [tex]\(4 - 5\sqrt{5}\)[/tex].
3. Multiply the Numerator and Denominator by the Conjugate:
[tex]\[ \frac{(2 + 3\sqrt{5})(4 - 5\sqrt{5})}{(4 + 5\sqrt{5})(4 - 5\sqrt{5})} \][/tex]
4. Simplify the Denominator:
Use the difference of squares formula: [tex]\((a + b)(a - b) = a^2 - b^2\)[/tex].
[tex]\[ (4 + 5\sqrt{5})(4 - 5\sqrt{5}) = 4^2 - (5\sqrt{5})^2 = 16 - 125 = -109 \][/tex]
5. Expand and Simplify the Numerator:
[tex]\[ (2 + 3\sqrt{5})(4 - 5\sqrt{5}) = 2 \cdot 4 + 2 \cdot (-5\sqrt{5}) + 3\sqrt{5} \cdot 4 + 3\sqrt{5} \cdot (-5\sqrt{5}) \][/tex]
Simplify each term:
[tex]\[ 2 \cdot 4 = 8, \quad 2 \cdot (-5\sqrt{5}) = -10\sqrt{5}, \quad 3\sqrt{5} \cdot 4 = 12\sqrt{5}, \quad 3\sqrt{5} \cdot (-5\sqrt{5}) = -75 \cdot 5 = -75 \][/tex]
Combine these terms:
[tex]\[ 8 - 75 - 10\sqrt{5} + 12\sqrt{5} \][/tex]
Simplify further:
[tex]\[ -67 + 2\sqrt{5} \][/tex]
6. Form the Fraction:
[tex]\[ \frac{-67 + 2\sqrt{5}}{-109} \][/tex]
7. Separate the Rational and Irrational Parts:
[tex]\[ \frac{-67}{-109} + \frac{2\sqrt{5}}{-109} \][/tex]
Simplify the fractions:
[tex]\[ \frac{-67}{-109} = \frac{67}{109} \quad \text{and} \quad \frac{2\sqrt{5}}{-109} = -\frac{2\sqrt{5}}{109} \][/tex]
Thus, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[ a = \frac{67}{109} \approx 0.6146788990825688, \quad b = -\frac{2}{109} \approx -0.01834862385321101 \][/tex]
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