Answers:
- a) [tex]7^{1/5}[/tex]
- b) [tex]7^{-1/7}[/tex]
Each answer has 7 as the base and a fraction as an exponent.
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Explanation:
One useful rule here is that [tex]\sqrt[n]{x^m} = x^{m/n}[/tex]. The fraction m/n is the exponent for the base x. Note how the index of the root (n) becomes the denominator of the fraction m/n. So for example if we had n = 3, then we'd be dealing with a cube root.
For part a), we have x = 7, m = 1 and n = 5 to get
[tex]\sqrt[n]{x^m}=x^{m/n}\\\\\sqrt[5]{7^1}=7^{1/5}\\\\\sqrt[5]{7}=7^{1/5}\\\\[/tex]
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Part b) is the same idea
We would find that
[tex]\sqrt[n]{x^m}=x^{m/n}\\\\\sqrt[7]{7^1}=7^{1/7}\\\\\sqrt[7]{7}=7^{1/7}\\\\[/tex]
Then apply the reciprocal to both sides. This is the same as raising both sides to the exponent -1
[tex]\sqrt[7]{7}=7^{1/7}\\\\\left(\sqrt[7]{7}\right)^{-1}=\left(7^{1/7}\right)^{-1}\\\\\frac{1}{\sqrt[7]{7}}=7^{1/7*(-1)}\\\\\frac{1}{\sqrt[7]{7}}=7^{-1/7}\\\\[/tex]
You could also use the rule that x^(-y) = 1/(x^y) when dealing with negative exponents.
In the second to last step shown above, I used the rule (x^y)^z = x^(y*z) on the right hand side.
