Question 1209835: Simplify 25^{\frac{1}{2} - \log 5 + \sqrt{3}}.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Let's simplify the expression step-by-step:
**1. Rewrite 25 as 5^2:**
25^(1/2 - log 5 + √3) = (5^2)^(1/2 - log 5 + √3)
**2. Apply the Power of a Power Rule:**
(5^2)^(1/2 - log 5 + √3) = 5^(2 * (1/2 - log 5 + √3))
**3. Distribute the 2:**
5^(2 * (1/2 - log 5 + √3)) = 5^(1 - 2log 5 + 2√3)
**4. Apply the Logarithm Power Rule (2log 5 = log 5^2):**
5^(1 - 2log 5 + 2√3) = 5^(1 - log 5^2 + 2√3) = 5^(1 - log 25 + 2√3)
**5. Separate the Exponents:**
5^(1 - log 25 + 2√3) = 5^1 * 5^(-log 25) * 5^(2√3)
**6. Apply the Logarithm Identity (a^(-log_a(x)) = 1/x):**
Remember that log 25 is log base 10 of 25. We can rewrite 5^(-log 25) using the change of base formula.
* 5^(-log(25)) = 5^(-log(25)/log(10))
* We can rewrite 5^(-log(25)) as 10^(log(5)*(-log(25)/log(10)))
* We can rewrite 5^(-log(25)) as 10^(-log(25)*log(5)/log(10))
* We can rewrite 5^(-log(25)) as 10^(-log(25)*log(5))
* We can rewrite 5^(-log(25)) as 10^(log(25^(-log(5))))
* We can rewrite 5^(-log(25)) as 25^(-log(5))
* We can rewrite 5^(-log(25)) as (5^2)^(-log(5))
* We can rewrite 5^(-log(25)) as 5^(-2log(5))
* We can rewrite 5^(-log(25)) as 5^(log(5^(-2)))
* We can rewrite 5^(-log(25)) as 5^(log(1/25))
* We can rewrite 5^(-log(25)) as 1/25
5^1 * 5^(-log 25) * 5^(2√3) = 5 * (1/25) * 5^(2√3)
**7. Simplify:**
5 * (1/25) * 5^(2√3) = (1/5) * 5^(2√3)
**Final Simplified Form:**
(1/5) * 5^(2√3) or 5^(2√3 - 1)
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