1) Change radical forms to fractional exponents using the rule:
The nth root of "a number" = "that number" raised to the reciprocal of n.
For example [tex] \sqrt[n]{3} = 3^{ \frac{1}{n} }[/tex].
The square root of 3 ([tex] \sqrt{3} [/tex]) = 3 to the one-half power ([tex]3^{ \frac{1}{2} }[/tex]).
The 5th root of 3 ([tex] \sqrt[5]{3} [/tex]) = 3 to the one-fifth power ([tex]3^{ \frac{1}{5} }[/tex]).
2) Now use the product of powers exponent rule to simplify:
This rule says [tex] a^{m} a^{n} = a^{m+n}
[/tex]. When two expressions with the same base (a, in this example) are multiplied, you
can add their exponents while keeping the same base.
You now have [tex](3^{ \frac{1}{2} })*(3^{ \frac{1}{5} })[/tex]. These two expressions have the same base, 3. That means you can add their exponents:
[tex](3^{ \frac{1}{2} })(3^{ \frac{1}{5} })\\
= 3^{(\frac{1}{2} + \frac{1}{5}) }\\
= 3^{\frac{7}{10}}[/tex]
3) You can leave it in the form [tex]3^{\frac{7}{10}}[/tex] or change it back into a radical [tex] \sqrt[10]{3^7} [/tex]
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Answer: [tex]3^{\frac{7}{10}}[/tex] or [tex] \sqrt[10]{3^7} [/tex]
