Question

find the cubic root of 96 in radicals

Answer 1

To find the cubic root of 96 in radicals, we can start by expressing 96 in terms of its prime factors.

The prime factorization of 96 is:
\[
96 = 2^5 \times 3^1
\]

Now we can express the cubic root of 96 using its prime factorization:
\[
\sqrt[3]{96} = \sqrt[3]{2^5 \times 3^1}
\]

Using the property of roots, we can separate the factors:
\[
\sqrt[3]{96} = \sqrt[3]{2^5} \times \sqrt[3]{3^1}
\]

Next, we simplify \(\sqrt[3]{2^5}\):
\[
\sqrt[3]{2^5} = \sqrt[3]{2^3 \times 2^2} = \sqrt[3]{2^3} \times \sqrt[3]{2^2} = 2 \times \sqrt[3]{2^2} = 2 \sqrt[3]{4}
\]

Now we have:
\[
\sqrt[3]{96} = 2 \sqrt[3]{4} \times \sqrt[3]{3} = 2 \sqrt[3]{12}
\]

Thus, the cubic root of 96 in radicals is:
\[
\sqrt[3]{96} = 2 \sqrt[3]{12}
\]