Sure, let's simplify the given expression step by step.
We begin with the expression:
[tex]\[
\sqrt{125} + \sqrt{36 + 64}
\][/tex]
### Step 1: Simplify Inside the Square Root
First, we simplify the expression inside the second square root:
[tex]\[
36 + 64 = 100
\][/tex]
Now the expression becomes:
[tex]\[
\sqrt{125} + \sqrt{100}
\][/tex]
### Step 2: Simplify the Square Roots
Next, we find the values of each square root.
1. For \(\sqrt{125}\):
125 can be factored into its prime factors:
[tex]\[
125 = 5 \times 25 = 5 \times (5 \times 5) = 5^3
\][/tex]
Taking the square root of 125:
[tex]\[
\sqrt{125} = \sqrt{5^3} = \sqrt{5^2 \times 5} = 5\sqrt{5}
\][/tex]
2. For \(\sqrt{100}\):
100 can also be factored:
[tex]\[
100 = 10 \times 10 = 10^2
\][/tex]
Taking the square root of 100:
[tex]\[
\sqrt{100} = 10
\][/tex]
### Step 3: Add the Simplified Square Roots
Now that we have simplified both square roots, we add them together:
[tex]\[
5\sqrt{5} + 10
\][/tex]
So, the simplified form of the expression \(\sqrt{125} + \sqrt{100}\) is:
[tex]\[
10 + 5\sqrt{5}
\][/tex]
### Conclusion
The simplified form of the expression \(\sqrt{125} + \sqrt{36 + 64}\) is:
[tex]\[
10 + 5\sqrt{5}
\][/tex]
