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Rewrite the expression in the form z^(n). Write the exponent as an integer, fraction, or an exact number). root(5)(z^(4)z^(-(3)/(2)))= ◻

Rewrite the expression in the form zn z^{n} .\newlineWrite the exponent as an integer, fraction, or an exact number).\newlinez4z−325= \sqrt[5]{z^{4} z^{-\frac{3}{2}}}= \newline□ \square

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Full solution

Q. Rewrite the expression in the form zn z^{n} .\newlineWrite the exponent as an integer, fraction, or an exact number).\newlinez4z−325= \sqrt[5]{z^{4} z^{-\frac{3}{2}}}= \newline□ \square
  1. Subtract Exponents: Now, we need to subtract the exponents.\newline4−32=4−1.5=2.54 - \frac{3}{2} = 4 - 1.5 = 2.5\newlineSo, z4−32=z2.5z^{4 - \frac{3}{2}} = z^{2.5}
  2. Express as Fraction: Next, we need to express the exponent 2.52.5 as a fraction for the fifth root.\newline2.52.5 can be written as 52\frac{5}{2}, so z2.5=z52z^{2.5} = z^{\frac{5}{2}}
  3. Apply Fifth Root: Now, we apply the fifth root to z5/2z^{5/2}. \newlinez5/25=z(52)(15)\sqrt[5]{z^{5/2}} = z^{\left(\frac{5}{2}\right)\left(\frac{1}{5}\right)}
  4. Multiply Exponents: Multiply the exponents (52)(\frac{5}{2}) by (15)(\frac{1}{5}).(52)×(15)=510=12(\frac{5}{2}) \times (\frac{1}{5}) = \frac{5}{10} = \frac{1}{2}So, z525=z12\sqrt[5]{z^{\frac{5}{2}}} = z^{\frac{1}{2}}

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