(p^1/5p^7/10p^1/2)/((p^8))^(-1/5)= square (Simplify your answer. Type exponential notation with positiv

To simplify the expression \(\frac {p^{1/5}p^{7/10}p^{1/2}}{(p^{8})^{-1/5}}\), we need to follow these steps:<br /><br />1. Combine the exponents in the numerator.<br />2. Simplify the denominator.<br />3. Divide the numerator by the denominator.<br /><br />Let's start with the numerator:<br />\[ p^{1/5} \cdot p^{7/10} \cdot p^{1/2} \]<br /><br />When multiplying terms with the same base, we add the exponents:<br />\[ p^{1/5 + 7/10 + 1/2} \]<br /><br />Convert all fractions to have a common denominator (which is 10 in this case):<br />\[ \frac{1}{5} = \frac{2}{10}, \quad \frac{1}{2} = \frac{5}{10} \]<br /><br />So,<br />\[ \frac{2}{10} + \frac{7}{10} + \frac{5}{10} = \frac{14}{10} = \frac{7}{5} \]<br /><br />Thus, the numerator simplifies to:<br />\[ p^{7/5} \]<br /><br />Now, let's simplify the denominator:<br />\[ (p^8)^{-1/5} \]<br /><br />Using the power rule \((a^m)^n = a^{mn}\):<br />\[ (p^8)^{-1/5} = p^{8 \cdot (-1/5)} = p^{-8/5} \]<br /><br />Now we have:<br />\[ \frac{p^{7/5}}{p^{-8/5}} \]<br /><br />When dividing terms with the same base, we subtract the exponents:<br />\[ p^{7/5 - (-8/5)} = p^{7/5 + 8/5} = p^{15/5} = p^3 \]<br /><br />Therefore, the simplified expression is:<br />\[ \boxed{p^3} \]