Question

find the cubic root of 189 in radicals

Answer 1

To express the cubic root of 189 in radicals, we can begin by factoring 189 into its prime factors.

1. First, we divide 189 by 3:
\[
189 \div 3 = 63
\]
2. Next, we divide 63 by 3:
\[
63 \div 3 = 21
\]
3. Then, we divide 21 by 3:
\[
21 \div 3 = 7
\]
4. Finally, we can't divide 7 by any prime numbers other than itself.

So, the prime factorization of 189 is:
\[
189 = 3^3 \times 7
\]

Now, we can find the cubic root of 189:
\[
\sqrt[3]{189} = \sqrt[3]{3^3 \times 7}
\]

Using the property of radicals, we can separate the factors:
\[
\sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7}
\]

Calculating the cubic root of \(3^3\):
\[
\sqrt[3]{3^3} = 3
\]

Therefore, we have:
\[
\sqrt[3]{189} = 3 \times \sqrt[3]{7}
\]

Thus, the cubic root of 189 in radicals is:
\[
\sqrt[3]{189} = 3\sqrt[3]{7}
\]