Certainly! Let's solve the expression step-by-step:
Expression to solve:
[tex]\[ \sqrt{2 + \frac{1}{2}} - \sqrt{5} \][/tex]
Step 1: Simplify the expression inside the first square root.
First, let's add the fractions inside the square root:
[tex]\[ 2 + \frac{1}{2} \][/tex]
Convert the whole number 2 into a fraction with a denominator of 2 to add it to [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ 2 = \frac{4}{2} \][/tex]
Now you can add the fractions:
[tex]\[ 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2} \][/tex]
Step 2: Take the square root of the resulting fraction.
Next, calculate the square root of [tex]\(\frac{5}{2}\)[/tex]:
[tex]\[ \sqrt{\frac{5}{2}} \][/tex]
The value of [tex]\(\sqrt{\frac{5}{2}}\)[/tex] is approximately [tex]\(1.5811388300841898\)[/tex].
Step 3: Evaluate the square root of 5.
Now, calculate the square root of 5:
[tex]\[ \sqrt{5} \][/tex]
The value of [tex]\(\sqrt{5}\)[/tex] is approximately [tex]\(2.23606797749979\)[/tex].
Step 4: Find the difference between the two square roots.
Finally, subtract [tex]\(\sqrt{5}\)[/tex] from [tex]\(\sqrt{\frac{5}{2}}\)[/tex]:
[tex]\[ \sqrt{\frac{5}{2}} - \sqrt{5} \approx 1.5811388300841898 - 2.23606797749979 \][/tex]
When you perform the subtraction, you get:
[tex]\[ 1.5811388300841898 - 2.23606797749979 \approx -0.6549291474156 \][/tex]
So, the final answer to the expression [tex]\(\sqrt{2+\frac{1}{2}} - \sqrt{5}\)[/tex] is approximately:
[tex]\[ \boxed{-0.6549291474156} \][/tex]
I hope this detailed explanation helps you understand the solution step-by-step! If you have any further questions, feel free to ask.
