Sure, let's go through each expression step-by-step and simplify these radicals.
1. Distribute and simplify: [tex]\(\sqrt{12} \cdot (-1 + \sqrt{5})\)[/tex]
[tex]\[
\sqrt{12} \cdot (-1 + \sqrt{5}) = \sqrt{12} \cdot (-1) + \sqrt{12} \cdot \sqrt{5}
\][/tex]
Simplify each term:
- [tex]\(\sqrt{12} \cdot (-1) = -\sqrt{12}\)[/tex]
- [tex]\(\sqrt{12} \cdot \sqrt{5} = \sqrt{12 \cdot 5} = \sqrt{60}\)[/tex]
Combining these:
[tex]\[
-\sqrt{12} + \sqrt{60}
\][/tex]
Numerically, this evaluates approximately to [tex]\(4.28186507727708\)[/tex].
2. Simplify: [tex]\(-4 \sqrt{3}\)[/tex]
This expression is already in its simplest form.
Numerically, this is approximately [tex]\(-6.928203230275509\)[/tex].
3. Simplify: [tex]\(-2 \sqrt{3} + 2 \sqrt{15}\)[/tex]
Each term is already simplified, so we present the expression as:
Numerically, this evaluates approximately to [tex]\(4.28186507727708\)[/tex].
4. Simplify: [tex]\(4 \sqrt{3}\)[/tex]
This expression is already in its simplest form.
Numerically, this is approximately [tex]\(6.928203230275509\)[/tex].
5. Simplify: [tex]\(6 \sqrt{3}\)[/tex]
This expression is already in its simplest form.
Numerically, this is approximately [tex]\(10.392304845413264\)[/tex].
These expressions, when simplified and calculated numerically, yield the values mentioned above.
