Let's simplify the given expression step by step.
First, consider the expression:
[tex]\[ 4 \sqrt{13} - 6 \sqrt{13} + 10 \sqrt{13} \][/tex]
### Step-by-Step Simplification:
1. Identify like terms: All the terms are multiples of [tex]\(\sqrt{13}\)[/tex]. Hence, we can factor [tex]\(\sqrt{13}\)[/tex] out of the expression:
[tex]\[ 4\sqrt{13} - 6\sqrt{13} + 10\sqrt{13} \][/tex]
2. Combine the coefficients: Add or subtract the coefficients of [tex]\(\sqrt{13}\)[/tex]:
[tex]\[ (4 - 6 + 10) \sqrt{13} \][/tex]
3. Compute the sum of the coefficients:
[tex]\[ 4 - 6 + 10 \][/tex]
First, simplify [tex]\(4 - 6\)[/tex]:
[tex]\[ 4 - 6 = -2 \][/tex]
Then add [tex]\(10\)[/tex]:
[tex]\[ -2 + 10 = 8 \][/tex]
So the combined coefficient is 8.
4. Substitute the combined coefficient back into the expression:
[tex]\[ 8 \sqrt{13} \][/tex]
Therefore, the simplified form of the expression [tex]\(4\sqrt{13} - 6\sqrt{13} + 10\sqrt{13}\)[/tex] is:
[tex]\[ 8 \sqrt{13} \][/tex]
### Verifying the Result Numerically:
1. Calculate each term individually:
[tex]\[
4\sqrt{13} \approx 14.422205101855958
\][/tex]
[tex]\[
-6\sqrt{13} \approx -21.633307652783935
\][/tex]
[tex]\[
10\sqrt{13} \approx 36.055512705569792
\][/tex]
2. Add these values together:
[tex]\[
14.422205101855958 - 21.633307652783935 + 36.055512705569792 = 28.844410203711917
\][/tex]
This numerical result approximates to the value of:
[tex]\[ 28.844410203711917 \][/tex]
3. Confirm [tex]\(\frac{28.844410203711917}{\sqrt{13}} \approx 8.000000000000002\)[/tex]:
[tex]\[
\frac{28.844410203711917}{\sqrt{13}} \approx 8.000000000000002
\][/tex]
This confirms that the expression simplifies to:
[tex]\[ 8\sqrt{13} \][/tex]
So, the correct simplified form of the expression is [tex]\(8 \sqrt{13}\)[/tex].
