Question
the simplified form of ∛54 is 3∛2 why is there a coefficient of 3 outside the cube root
Answers
GPT-4o mini
To simplify ∛54 (the cube root of 54), we can factor 54 to see if it has any perfect cubes.
The prime factorization of 54 is:
\[
54 = 27 \times 2 = 3^3 \times 2
\]
Since \(27\) is a perfect cube (\(3^3\)), we can simplify the cube root as follows:
\[
\sqrt[3]{54} = \sqrt[3]{27 \times 2} = \sqrt[3]{27} \times \sqrt[3]{2}
\]
Now, calculate the cube root of \(27\):
\[
\sqrt[3]{27} = 3
\]
So we have:
\[
\sqrt[3]{54} = 3 \times \sqrt[3]{2}
\]
Thus, the simplified form of \(\sqrt[3]{54}\) is:
\[
3\sqrt[3]{2}
\]
The coefficient of \(3\) outside the cube root comes from the fact that \(27\) is equal to \(3^3\), which allows us to take the cube root of \(27\) and obtain the \(3\) as a coefficient.
The prime factorization of 54 is:
\[
54 = 27 \times 2 = 3^3 \times 2
\]
Since \(27\) is a perfect cube (\(3^3\)), we can simplify the cube root as follows:
\[
\sqrt[3]{54} = \sqrt[3]{27 \times 2} = \sqrt[3]{27} \times \sqrt[3]{2}
\]
Now, calculate the cube root of \(27\):
\[
\sqrt[3]{27} = 3
\]
So we have:
\[
\sqrt[3]{54} = 3 \times \sqrt[3]{2}
\]
Thus, the simplified form of \(\sqrt[3]{54}\) is:
\[
3\sqrt[3]{2}
\]
The coefficient of \(3\) outside the cube root comes from the fact that \(27\) is equal to \(3^3\), which allows us to take the cube root of \(27\) and obtain the \(3\) as a coefficient.