Sure, let's simplify the expression step-by-step:
Expression:
[tex]\[ 4 \sqrt{96} \times \sqrt{50} \][/tex]
Step 1: Simplify the square roots individually
First, let's simplify [tex]\(\sqrt{96}\)[/tex] and [tex]\(\sqrt{50}\)[/tex].
1. [tex]\(\sqrt{96}\)[/tex]:
[tex]\[
96 = 16 \times 6 = 4^2 \times 6
\][/tex]
Using the property of square roots:
[tex]\[
\sqrt{96} = \sqrt{4^2 \times 6} = \sqrt{4^2} \times \sqrt{6} = 4 \sqrt{6}
\][/tex]
2. [tex]\(\sqrt{50}\)[/tex]:
[tex]\[
50 = 25 \times 2 = 5^2 \times 2
\][/tex]
Using the property of square roots:
[tex]\[
\sqrt{50} = \sqrt{5^2 \times 2} = \sqrt{5^2} \times \sqrt{2} = 5 \sqrt{2}
\][/tex]
Step 2: Substitute the simplified square roots back into the expression
Now, our expression becomes:
[tex]\[
4 \sqrt{96} \times \sqrt{50} = 4 \times (4 \sqrt{6}) \times (5 \sqrt{2})
\][/tex]
Simplify inside the multiplication:
[tex]\[
= 4 \times 4 \times 5 \times \sqrt{6} \times \sqrt{2}
\][/tex]
Step 3: Simplify the constants and the radical terms
Combine the constants:
[tex]\[
4 \times 4 \times 5 = 80
\][/tex]
Combine the radical terms using the property [tex]\(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\)[/tex]:
[tex]\[
\sqrt{6} \times \sqrt{2} = \sqrt{6 \times 2} = \sqrt{12}
\][/tex]
So the expression now looks like:
[tex]\[
80 \times \sqrt{12}
\][/tex]
Step 4: Simplify [tex]\(\sqrt{12}\)[/tex]
Simplify [tex]\(\sqrt{12}\)[/tex]:
[tex]\[
12 = 4 \times 3 = 2^2 \times 3
\][/tex]
[tex]\[
\sqrt{12} = \sqrt{2^2 \times 3} = \sqrt{2^2} \times \sqrt{3} = 2 \sqrt{3}
\][/tex]
Substitute this back into the expression:
[tex]\[
80 \times 2 \sqrt{3} = 160 \sqrt{3}
\][/tex]
Hence the simplified form of [tex]\(4 \sqrt{96} \times \sqrt{50}\)[/tex] is:
[tex]\[
160 \sqrt{3}
\][/tex]
Finally, [tex]\(\approx 277.12812921102034\)[/tex].
