Sure, let's break down the given expression step-by-step.
The expression we need to simplify and solve is:
[tex]\[ -5 \sqrt{10} - \frac{3}{2} + \pi + \sqrt[3]{27} \][/tex]
### Step 1: Evaluate each term separately
1. First Term:
[tex]\[ -5 \sqrt{10} \][/tex]
This involves multiplying -5 by the square root of 10. The square root of 10 is an irrational number approximately equal to [tex]\( 3.162277660168379 \)[/tex]. Thus:
[tex]\[ -5 \times 3.162277660168379 \approx -15.811388300841898 \][/tex]
2. Second Term:
[tex]\[ -\frac{3}{2} \][/tex]
This is a straightforward fraction. Converting it to a decimal gives:
[tex]\[ -\frac{3}{2} = -1.5 \][/tex]
3. Third Term:
[tex]\[ \pi \][/tex]
The well-known mathematical constant [tex]\( \pi \)[/tex]. It is approximately equal to:
[tex]\[ \pi \approx 3.141592653589793 \][/tex]
4. Fourth Term:
[tex]\[ \sqrt[3]{27} \][/tex]
This term means the cube root of 27. Since [tex]\( 3^3 = 27 \)[/tex], we get:
[tex]\[ \sqrt[3]{27} = 3 \][/tex]
### Step 2: Combine the evaluated terms
Now, let's sum up all the terms we have calculated:
[tex]\[ -15.811388300841898 - 1.5 + 3.141592653589793 + 3 \][/tex]
Performing the addition and subtraction step-by-step:
1. Adding the negative terms first:
[tex]\[ -15.811388300841898 - 1.5 \approx -17.311388300841898 \][/tex]
2. Adding the positives:
[tex]\[ 3.141592653589793 + 3 \approx 6.141592653589793 \][/tex]
3. Summing these two results:
[tex]\[ -17.311388300841898 + 6.141592653589793 \approx -11.169795647252105 \][/tex]
### Final Answer:
Thus, the value of the given expression:
[tex]\[ -5 \sqrt{10} - \frac{3}{2} + \pi + \sqrt[3]{27} \][/tex]
is approximately:
[tex]\[ -11.169795647252105 \][/tex]
