Sure, let's solve each part of the problem step-by-step:
Part 1: Calculate the cube root of [tex]\(216 \times 125000\)[/tex]
1. First, find the product of 216 and 125000:
[tex]\[
216 \times 125000 = 27000000
\][/tex]
2. Next, find the cube root of 27000000:
[tex]\[
\sqrt[3]{27000000} \approx 299.9999999999999
\][/tex]
Therefore, the cube root of [tex]\(216 \times 125000\)[/tex] is approximately [tex]\(299.9999999999999\)[/tex].
Part 2: Calculate the cube root of [tex]\(-343 \times 64\)[/tex]
1. First, find the product of -343 and 64:
[tex]\[
-343 \times 64 = -21952
\][/tex]
2. Next, find the cube root of -21952. This involves dealing with complex numbers, as the cube root of a negative number is not a real number:
[tex]\[
\sqrt[3]{-21952} = 14.000000000000002 + 24.248711305964278j
\][/tex]
Therefore, the cube root of [tex]\(-343 \times 64\)[/tex] is approximately [tex]\(14.000000000000002 + 24.248711305964278j\)[/tex].
In summary:
1. The cube root of [tex]\(216 \times 125000\)[/tex] is approximately [tex]\(299.9999999999999\)[/tex].
2. The cube root of [tex]\(-343 \times 64\)[/tex] is approximately [tex]\(14.000000000000002 + 24.248711305964278j\)[/tex].
