Evaluate the expression without using a calculator. Unless stated otherwise, all letters represent positive numbers. e^(ln√x+5)
To evaluate the expression e^(ln√x+5) without using a calculator, we need to understand the properties of the natural logarithm (ln) and the exponential function (e).
First, let's break down the expression step by step:
1. Start with the innermost part, ln√x. The square root (√) of a positive number x can be expressed as x^(1/2), so ln√x becomes ln(x^(1/2)).
2. The natural logarithm (ln) and exponential function (e) are inverse functions of each other. This means that ln(e^a) = a, and e^(lna) = a for any positive number a.
3. Using the inverse property, the expression ln(x^(1/2)) simplifies to (1/2)*ln(x).
4. Next, we add 5 to the simplified expression: (1/2)*ln(x) + 5.
5. Finally, we apply the exponential function e to the whole expression, e^((1/2)*ln(x) + 5).
Now, let's evaluate the expression using these steps:
1. e^((1/2)*ln(x) + 5).
Note: Since the prompt states that all letters represent positive numbers, we do not have an actual value for x. Therefore, we can only give a general representation of the evaluated expression.
2. We can simplify the exponent by applying the rules of exponents:
e^((1/2)*ln(x) + 5) = e^((1/2)*ln(x)) * e^5.
3. Applying the properties of the exponential function, e^((1/2)*ln(x)) = x^(1/2):
e^((1/2)*ln(x)) * e^5 = x^(1/2) * e^5.
So, the evaluated expression without using a calculator is x^(1/2) * e^5.
Note: If you have the actual value of x, you can substitute it into the expression x^(1/2) * e^5 to calculate the final result.