Question
how can re write a cube root of 3√5 using a rational exponent
Answers
Answered by
GPT-4o mini
The cube root of a number can be expressed using a rational exponent. In this case, the cube root of \(5\) can be written as:
\[
\sqrt[3]{5} = 5^{1/3}
\]
Therefore, you can rewrite the cube root of \( \sqrt{5} \) as:
\[
\sqrt[3]{\sqrt{5}} = (\sqrt{5})^{1/3}
\]
Since the square root of \(5\) can also be expressed using a rational exponent (as \(5^{1/2}\)), we can further simplify it:
\[
(\sqrt{5})^{1/3} = (5^{1/2})^{1/3}
\]
Now, using the property of exponents \((a^m)^n = a^{m \cdot n}\), we get:
\[
(5^{1/2})^{1/3} = 5^{(1/2) \cdot (1/3)} = 5^{1/6}
\]
So, the cube root of \(\sqrt{5}\) can be written as:
\[
\sqrt[3]{\sqrt{5}} = 5^{1/6}
\]
\[
\sqrt[3]{5} = 5^{1/3}
\]
Therefore, you can rewrite the cube root of \( \sqrt{5} \) as:
\[
\sqrt[3]{\sqrt{5}} = (\sqrt{5})^{1/3}
\]
Since the square root of \(5\) can also be expressed using a rational exponent (as \(5^{1/2}\)), we can further simplify it:
\[
(\sqrt{5})^{1/3} = (5^{1/2})^{1/3}
\]
Now, using the property of exponents \((a^m)^n = a^{m \cdot n}\), we get:
\[
(5^{1/2})^{1/3} = 5^{(1/2) \cdot (1/3)} = 5^{1/6}
\]
So, the cube root of \(\sqrt{5}\) can be written as:
\[
\sqrt[3]{\sqrt{5}} = 5^{1/6}
\]