Question
Find the cube root of 81: 81−−√3 .
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Answered by
GPT-4o mini
To find the cube root of 81, you can express it as \( 81^{1/3} \).
1. First, write 81 in terms of its prime factors:
\[
81 = 3^4
\]
2. Now, apply the cube root:
\[
(81)^{1/3} = (3^4)^{1/3} = 3^{4/3}
\]
3. This can be expressed as:
\[
3^{4/3} = 3^{1 + 1/3} = 3 \cdot 3^{1/3}
\]
4. Finally, \( 3^{1/3} \) is the cube root of 3, so the final expression for the cube root of 81 is:
\[
\sqrt[3]{81} = 3 \cdot \sqrt[3]{3}
\]
Thus, the cube root of 81 is \( 3 \cdot \sqrt[3]{3} \).
1. First, write 81 in terms of its prime factors:
\[
81 = 3^4
\]
2. Now, apply the cube root:
\[
(81)^{1/3} = (3^4)^{1/3} = 3^{4/3}
\]
3. This can be expressed as:
\[
3^{4/3} = 3^{1 + 1/3} = 3 \cdot 3^{1/3}
\]
4. Finally, \( 3^{1/3} \) is the cube root of 3, so the final expression for the cube root of 81 is:
\[
\sqrt[3]{81} = 3 \cdot \sqrt[3]{3}
\]
Thus, the cube root of 81 is \( 3 \cdot \sqrt[3]{3} \).
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