Question 1209755: If x = [(a + √(a² - 1)]^(2mn/(m-n)),
Then [x^(1/m) + x^(1/n)]²/[x^[(1/m)+(1/n)]] = 1444,
find a.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step:
**1. Simplify the Expression:**
Let's simplify the expression:
[x^(1/m) + x^(1/n)]² / x^[(1/m) + (1/n)]
= [x^(1/m) + x^(1/n)]² / x^[(m+n)/mn]
= [x^(1/m)]² + 2x^(1/m)x^(1/n) + [x^(1/n)]² / x^[(m+n)/mn]
= [x^(2/m) + 2x^((1/m)+(1/n)) + x^(2/n)] / x^[(m+n)/mn]
= x^(2/m) / x^[(m+n)/mn] + 2x^((1/m)+(1/n)) / x^[(m+n)/mn] + x^(2/n) / x^[(m+n)/mn]
= x^(2/m - (m+n)/mn) + 2x^((1/m)+(1/n) - (m+n)/mn) + x^(2/n - (m+n)/mn)
= x^((2n - m - n)/mn) + 2x^((mn(1/m+1/n) - m - n)/mn) + x^((2m - m - n)/mn)
= x^((n - m)/mn) + 2x^((m+n - m - n)/mn) + x^((m - n)/mn)
= x^((n - m)/mn) + 2x^0 + x^((m - n)/mn)
= x^((n - m)/mn) + 2 + x^((m - n)/mn)
= x^((n - m)/mn) + 2 + x^(-(n - m)/mn)
Let y = (n - m)/mn. Then the expression becomes:
x^y + 2 + x^(-y)
**2. Substitute the Value of x:**
Given x = [(a + √(a² - 1)]^(2mn/(m-n)), we have:
x^y = [(a + √(a² - 1)]^(2mn/(m-n)) * (n-m)/mn = (a + √(a² - 1))^(-2)
x^(-y) = [(a + √(a² - 1)]^(2mn/(m-n)) * (m-n)/mn = (a + √(a² - 1))^(2)
Therefore, the expression becomes:
(a + √(a² - 1))^(-2) + 2 + (a + √(a² - 1))^(2) = 1444
**3. Simplify Further:**
Let z = a + √(a² - 1). Then:
1/z² + 2 + z² = 1444
z² + 1/z² = 1442
(z + 1/z)² = z² + 2 + 1/z² = 1444
z + 1/z = ±38
**4. Solve for z:**
z + 1/z = 38 (we can ignore the negative solution as z is positive)
z² - 38z + 1 = 0
Using the quadratic formula:
z = [38 ± √(38² - 4)] / 2
z = [38 ± √1440] / 2
z = 19 ± √360
z = 19 ± 6√10
Since z = a + √(a² - 1), we have:
a + √(a² - 1) = 19 + 6√10
a + √(a² - 1) = (3+√10)²
a + √(a² - 1) = 3² + 2*3*√10 + (√10)²
a + √(a² - 1) = 9 + 6√10 + 10
a + √(a² - 1) = 19 + 6√10
Therefore a = 19.
**5. Verification:**
If a = 19, then √(a² - 1) = √(19² - 1) = √360 = 6√10
a + √(a² - 1) = 19 + 6√10
**Final Answer:**
a = 19
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